Criteria for Solid Tumor Expansion and Lymphoma Cohorts:\r\n* Inclusion and exclusion criteria for this cohort are the same as above, with the following rule for CD4 count based on tolerability in Phase I; if, participants with lymphocyte T CD4 count between 100-200/mm^3 (Stratum 2) are shown to tolerate treatment in the Phase I dose de-escalation portion at the same dose level as those with CD4 counts > 200/mm^3 (Stratum 1), participants in the expansion cohort with CD4 counts >= 100/mm^3 are permitted; otherwise, the expansion is open to all solid tumor patients except those whose tumors are known not to respond to nivolumab (pancreas, prostate and MSS colon cancer); for the relapsed refractory HIV-cHL cohort, participants with CD4 count >= 100/mm^3 are permitted CD4 count >= 50 cells/ul Patients who have BCR-ABL fusion transcript determined by fluorescence in situ hybridization (FISH) or real time-polymerase chain reaction (RT-PCR) or t(9;22)(q34;q11) by cytogenetics are not eligible and should be considered for enrollment on studies that incorporate imatinib during induction; please note: patients must also be assessed for CD20 positivity and other markers; positivity for CD22 and CD20 is defined as baseline expression of the CD22 or CD20 antigen in more than 20% of leukemic cells using local multiparameter flow-cytometric immunophenotyping with the use of CD45 expression as a marker to gate the ALL blast population, according to recommendations from the European LeukemiaNet Participants must have a CD4 count performed within 30 days of enrollment ELIGIBILITY CRITERIA - PHASE II (ARM D): Patient cannot have poorly controlled HIV, or CD4 < 400; HIV positive patients are allowed on this study if they have a CD4 count >= 400, and are on a stable antiviral regimen Dose Expansion Cohort #1 - Patients will have relapse of CD123+ BPDCN. Patients with prior CD123-targeting agents will be allowed as long as the blasts still have detectable CD123 expression. Histologically documented CD20-positive lymphoma Known CD20-negative status at relapse or progression Prior treatment with CD47 or signal regulatory protein alpha (SIRP?) targeting agents. Clinical and phenotypic verification of B cell CLL/ SLL/ or monoclonal B-cell lymphocytosis (MBL) and measurable disease; immunophenotyping of the leukemic cells (blood, lymph node, or bone marrow) must demonstrate a monoclonal (kappa or lambda light chain restricted) B cell population with immunophenotype diagnostic for CLL (e.g.: co-expressing CD19 and CD5) Histologically documented CD20-positive B-cell lymphoma as determined by the local laboratory Known CD20-negative status at relapse or progression Prior treatment with the following agents (from last dose of prior treatment to first dose of GSK3174998): Tumor necrosis factor receptor (TNFR) agonists, including OX40, CD27, CD137 (4-1BB), CD357 (GITR): at any time; Checkpoint inhibitors, including PD-1, PD-L1, and anti-cytotoxic T-lymphocyte antigen 4 (CTLA-4) inhibitors: within 4 weeks. ADJUVANT COHORT: Derived benefit from PD 0332991 in the neoadjuvant setting in this trial; this includes the 26 patients who achieved complete cell cycle arrest only after the addition of PD 0332991 (C1D1 Ki67 > 2.7% and C1D15 Ki67 =< 2.7%) from the main study (PIK3CA WT, mutant, or unknown cohorts) as well as any patients who have a Ki67 =< 10% on C1D15 biopsy in the endocrine resistant cohort Patient with C-kit (CD117) positive tumour detected immuno-histochemically Obtained within 14 days prior to C1D1: HCV NA analysis negative Subjects who have undergone prior anti-CD19 or anti-CD22 CAR therapy will be eligible if < 5% of circulating levels of CD3+ cells express the previous CAR by flow cytometry >= 1% of CD30-positivity by immunohistochemistry confirmed by hematopathology review at the participating institution Prior treatment with a CD47 or signal regulatory protein (SIRP) alpha targeting agent. Must have received adequate prior therapy for the underlying CD20+ B-cell lymphoma, defined as an anti-CD20 mAb in combination with an anthracycline-containing chemotherapy regimen (i.e. chemo-immunotherapy) and at least one of the following: Known CD20-negative status at relapse or progression Residual disease at the time of transplant or post-transplant relapse is defined as polymerase chain reaction (PCR) positivity, specific cytogenetic abnormalities, an abnormal population on flow cytometry, or increased blasts on bone marrow biopsy or in the peripheral blood; minimal residual disease (MRD) will be defined as detection in blood or marrow of any of the following:\r\n* Any leukemia-specific marker (such as t(12;21); t(9;22) or t(4;11)) documented in the patient’s leukemia cells pre-transplant on a post-transplant evaluation\r\n* A leukemia-specific phenotype (e.g. expression of markers including CD10 and/or CD19 or CD3 and/or CD4 or CD8) post-transplant at a level of ? 0.01%\r\n* Mixed donor chimerism (> 20%) Presence of lymphadenopathy and/or splenomegaly with histopathological evaluation of a lymph node biopsy consistent with CLL. Clonality of the circulating B-lymphocytes should be confirmed at screening. Previously confirmed immunohistological diagnosis with a characteristic CD5+/CD20+ B--cell immunophenotype according to WHO criteria. CLL diagnosis requires an absolute peripheral blood monoclonal CD20+/CD5+ B-lymphocyte count ? 5000 cells/?L for the duration of at least 3 months. Part 2: Diagnosis of CLL according to the National Cancer Institute (NCI) criteria or SLL according to the World Health Organization (WHO) criteria; this includes previous documentation of:\r\n* Biopsy-proven small lymphocytic lymphoma or\r\n* Diagnosis of CLL according to NCI working group criteria as evidenced by all of the following:\r\n** Peripheral blood B cell count of > 5 x 10^9/L consisting of small to moderate size lymphocytes\r\n** Immunophenotyping consistent with CLL defined as:\r\n*** The predominant population of lymphocytes share both B-cell antigens (cluster of differentiation [CD]19, CD20 [typically dim expression] or CD23) as well as CD5 in the absence of other pan-T-cell markers (CD3, CD2, etc.)\r\n*** Clonality as evidenced by kappa or lambda light chain expression (typically dim immunoglobulin expression) or other genetic method (e.g. immunoglobulin heavy chain variable [IGHV] analysis)\r\n*** NOTE: splenomegaly, hepatomegaly, or lymphadenopathy are not required for the diagnosis of CLL\r\n** Before diagnosing CLL or SLL, mantle cell lymphoma must be excluded by demonstrating a negative fluorescent in situ hybridization (FISH) analysis for t(11;14)(immunoglobulin H [IgH]/cyclin D1 [CCND1]) on peripheral blood or tissue biopsy or negative immunohistochemical stains for cyclin D1 on involved tissue biopsy Clinical and phenotypic verification of B cell CLL and measurable disease; immunophenotyping of the leukemic cells (blood or marrow) must demonstrate a monoclonal (or light chain positive) with immunophenotype consistent with B cell population (e.g., co-expressing cluster of differentiation [CD]19 and CD5) CD19+ leukemia RETREATMENT WITH MODIFIED T-CELLS INCLUSION CRITERIA: Subject has evidence of persistence of leukemic cells OR has CD19+ B cell recovery detected within 1 year of initial T cell infusion Patients must have measurable or evaluable disease; patients with > 10% of the peripheral blood mononuclear cells (PBMCs) having the characteristic abnormal (i.e., cluster of differentiation [CD]3dim, CD4+ CD25+ expressing) fluorescence-activated cell sorting (FACS) profile for circulating ATL cells will be considered to have evaluable disease Patients must have histologically confirmed B-lineage acute lymphoblastic leukemia (ALL) at diagnosis and either evidence of relapse/refractory disease based on a bone marrow/peripheral blood examination or evidence by cytogenetic studies or polymerase chain reaction (PCR) amplification; patients with only extramedullary disease in the absence of bone marrow or blood involvement are not eligible; patients with L3 (Burkitt's) are not eligible; for ALL in marrow or peripheral blood, immunophenotyping of the blood or marrow lymphoblasts must be performed to determine lineage (B cell, T-cell, or mixed B/T cell); NOTE: appropriate marker studies including CD19 (B cell), CD10, CD5, and CD7 (T cell) must be performed; co-expression of myeloid antigens (CD13 and CD33) will not exclude patients; if possible, the lineage specific markers cytoplasmic CD22 or CD79a (B cells), cytoplasmic CD3 (T cells) and cytoplasmic myeloperoxidase (MPO) (myeloid cells) must be determined; patients with mixed lineage ALL (ML-ALL) as defined by a lack of cytochemical markers of myeloid differentiation, and by the presence of immunophenotypic markers suggesting both lymphoid and myeloid differentiation, are allowed CD19 and/or CD22 must be expressed on at least 50% of the lymphoblasts Subjects must be diagnosed with CLL/SLL (chronic lymphocytic leukemia/small lymphocytic lymphoma) based on the standard histologic and immunophenotypic criteria described in the World Health Organization (WHO) classification of lymphoid malignancies, including immunophenotypic confirmation that the tumor cells co-express B cell antigens cluster of differentiation (CD)19/20 and CD5; mantle cell lymphoma should be excluded based on positive staining of the tumor cells for CD23, or the absence of staining of the tumor cells for cyclin D1 or the absence of t(11;14); this diagnosis should be confirmed at a Dana-Farber Harvard Cancer Center institution (Dana-Farber Cancer Institute [DFCI], Brigham and Women's Hospital [BWH], Massachusetts General Hospital [MGH], Beth Israel Deaconess Medical Center [BIDMC]) within approximately one month after the subject is registered; any question on histology confirmation should be brought to the attention of the Principal Investigator Evidence of HCL by flow cytometry, reviewed by the Laboratory of Pathology, National Cancer Institute (NCI), including positivity for CD19, CD22, CD20, and CD11c TREATMENT WITH SJCAR19: Available SJCAR19 product with >= 15% expression of the CD19-chimeric antigen receptor (CAR), and killing of CD19+ targets >= 20% in an in vitro cytotoxicity assay Prior treatment with a therapeutic agent targeting CD33 and/or CD3 (e.g. gemtuzumab ozogamicin, SGN-CD33A or AMG 330). Radiotherapy within the last 21 days prior to C1D1 (limited field palliative radiation is allowed if ? 14 days prior to C1D1); Prior treatment with CD74 targeting therapy Clinical and phenotypic verification of B cell CLL or SLL and measurable disease. Immunophenotyping of the leukemic cells (blood, lymph node, or bone marrow) must demonstrate a monoclonal (or light chain positive) B cell population with immunophenotype consistent with CLL (e.g.: co-expressing CD19, CD5, and CD23) Patients with HCL variant (as defined by absence of expression of CD25) Prior CD19 directed therapy other than blinatumomab Prior treatment with blinatumomab or CD-directed CAR T-cell therapy Evidence of at least one measurable lesion on imaging with the following exceptions\r\n* Patients treated with interim chemotherapy for disease control between enrollment and ET190L1-ARTEMIS T cell infusion who do not have measurable disease at re-screening are still eligible\r\n* CLL/SLL with documented B-cell absolute lymphocytosis > 5 x 10^9 cells/L peripheral blood or infiltration of lymph nodes and/or bone marrow infiltration by CLL phenotype cells defined as clonal B cells with majority population co-expressing CD5 and CD19, with surface immunoglobulin (sIg, kappa or lambda but not both) and CD20 (dim), CD23+ (if CD20 or sIg are bright or if CD23 is dim or negative [atypical CLL phenotype] then fluorescence in situ hybridization (FISH) for 11:14 translocation must be performed to differentiate from mantle cell lymphoma)\r\n* Lesions previously irradiated will be considered measurable only if progression has been documented following completion of radiation therapy\r\n** Measurable lesions are nodes/nodal masses > 1.5 cm, extranodal masses > 1.0 cm or positron emission tomography (PET) avid lesions consistent with lymphoma CD19+22+ leukemia Subjects must NOT have an active malignancy other than CD19+CD22+ leukemia CD4 count > 200/uL Previous treatment with CD19-targeted CAR T cells Subjects who have undergone prior anti-CD19 or anti-CD22 CAR therapy will be eligible if < 5% of circulating levels of CD3+ cells express the previous CAR by flow cytometry Documentation of CD20+ status Histologically confirmed CD30-positive lymphoma with cutaneous presentation biopsy-confirmed CD20+ expression of the underlying malignancy with disease progression following immediate prior therapy must have received adequate prior therapy for the underlying CD20+ B-cell lymphoma, defined as an anti-CD20 mAb in combination with an anthracycline-containing chemotherapy regimen (i.e. chemo-immunotherapy) and at least one of the following: Prior to lymphodepletion on protocol PLAT-02, total CD19 antigen load in the bone marrow is <15% of cells analyzed by flow cytometry (CD19 antigen load includes both CD19+ leukemia cells and non-malignant B cells) Patients not in remission must have CD45-expressing leukemic blasts; patients in remission do not require phenotyping and may have leukemia previously documented to be CD45 negative Documentation of CD19 expression on any prior or current tumor biopsy; patients who have received previous CD19-targeted therapy must have CD19-positive disease confirmed on a biopsy since completing the prior CD19-targeted therapy CRITERIA FOR SCREENING: For patients in stage 1 only, prior treatment with any CD19 CAR T-cell therapy is excluded Clear CD30 expression must be detected on 75% or more of malignant cells from either bone marrow or lymphoma mass by flow cytometry or immunohistochemistry; the patient’s malignancy will need to be assessed for CD30 expression by flow cytometry or immunohistochemistry performed at the National Institutes of Health (NIH); if unstained, paraffin-embedded bone marrow or lymphoma sections are available from prior biopsies, these can be used to determine CD30 expression by immunohistochemistry; otherwise, patients will need to come to the NIH for a biopsy to determine CD30 expression; the sample for CD30 expression can come from a biopsy obtained at any time before enrollment, unless the patient has received a prior anti-CD30 monoclonal antibody, in which case the sample must come from a biopsy following completion of the most recent anti-CD30 monoclonal antibody treatment Documented CD20+ FL. CD4 count >= 100 in the dose-finding cohort; once the dose-finding cohort is complete and if safety is established, participants with any CD4 count, including CD4 count < 100, will be allowed in the dose-escalation phase Participants who have received previous CD19-targeted therapy must have CD19-positive lymphoma confirmed on a biopsy since completing the prior CD19-targeted therapy. Pathologically confirmed MCL, with documentation of monoclonal CD20+ B cells that have a chromosome translocation t(11;14)(q13;q32) and/or overexpress cyclin D1. Subjects who had a thromboembolic event ? 4 weeks of C1D1 must be receiving adequate anticoagulation treatment for at least 2 weeks before C1D1 and must continue as clinically indicated post first dose Positive CD30+ immunohistochemical expression Sufficient baseline tumor tissue available to perform tumor infiltrating CD8+ T-cell density assessment; (NOTE: The actual CD8+ T-cell density assessment does not need to be performed prior to registration, however the slides must be reviewed prior to registration to ensure that adequate tissue will be available for the pre-treatment tumor infiltrating CD8+ T-cell density assessment) Measurable unresectable cancer expressing CD70 as assessed by immunohistochemistry of resected tissue (>= 2+ CD70 positive on >= 50% of cancer cells, or >= 1+ CD70 positive on >= 75% of cancer cells) ELIGIBILITY FOR TREATMENT WITH TCRC4-TRANSDUCED CD8+ CELLS EXCLUSION FOR TREATMENT WITH TCRC4-TRANSDUCED CD8+ CELLS CD19 negative B-cell leukaemia Subjects must have a diagnosis of relapsed or refractory mantle cell lymphoma as follows:\r\n* Diagnosis of mantle cell lymphoma (MCL) must include morphology and expression of either cyclin D1 in association with other relevant markers (eg, CD19, CD20, CD5) or evidence of t(11;14) as assessed by cytogenetics, fluorescent in situ hybridization (FISH), or polymerase chain reaction (PCR)\r\n* Relapsed or refractory disease is defined as no response or progressive disease to prior treatment if the prior treatment comprised any of the following:\r\n** 1 regimen containing an anti-CD20 antibody administered for >= 2 doses, and/or\r\n** >= 1 regimen containing 1 cytotoxic agent (e.g., bendamustine, chlorambucil, cyclophosphamide, cytarabine, doxorubicin) administered for 2 cycles Prior treatment with any CD19 CAR T-cell therapy Prior treatment with programmed cell death (PD)-1, PD-ligand (L)1, cytotoxic T lymphocyte-associated protein 4 (CTLA 4) targeted therapy, or tumor necrosis factor receptor superfamily (TNFRSF) agonists including CD134 (OX40), CD27, CD137 (4-1BB), and CD357 (glucocorticoid-induced tumor necrosis factor receptor family-related protein [GITR]) Evidence of CD19 expression on any prior or current tumor specimen or a high likelihood of CD19 expression based on disease histology Documentation of CD19 expression on any prior or current tumor biopsy PRIOR TO CELL PROCUREMENT: CD30+ disease (result can be pending at the time of cell procurement, but must be confirmed prior to treatment with ATLCAR.CD30 cells); NOTE: CD30+ disease requires documented CD30 expression by immunohistochemistry based on the institutional hematopathology standard PRIOR TO LYMPHODEPLETION: CD30+ disease (result can be pending at the time of cell procurement, but must be confirmed prior to lymphodepletion); NOTE: CD30+ disease requires documented CD30 expression by immunohistochemistry based on the institutional hematopathology standard PRIOR TO INFUSION OF ATLCAR.CD30 CELLS: AST ? 3 times ULN PRIOR TO INFUSION OF ATLCAR.CD30 CELLS: Serum creatinine ? 1.5 times ULN Patients must have measurable or evaluable disease; NOTE: All patients with greater than 10% abnormal CD4+ homogeneous CD3^low strongly CD25+ expressing cells, or greater than 5% Sezary/T-PLL cells, among the peripheral blood mononuclear cells (PBMCs) in the peripheral blood will be deemed to have evaluable disease CD4 count >= 100 cells/uL if HIV-infected Exposure to another CD37 targeting drug. Documentation of CD22 expression on malignant cells at relapse PATIENT CRITERIA FOR PROCEEDING WITH CD8+ MEMORY T-CELL INFUSION: Patients must have evidence of mixed CD3 T-cell chimerism based on the day +28 (+/- 7 days) blood sample showing >= 30% and =< 90% donor type cells Diagnosis of CLL: monoclonal B-cells co-expressing >= one B-cell marker (cluster of differentiation [CD]19, CD20, or CD23) and CD5 in peripheral blood or lymph node Expression of CD-22 in >= 20% blasts DONOR: Cell yield goal (post CD34 determination): > 1-6 x10^6 CD34+ cells/kg recipient CD19-positive tumor Patients should have a confirmed diagnosis of chronic lymphocytic leukemia defined as all of the following:\r\n* Absolute lymphocyte count (ALC) > 5000\r\n* Positive for either CD19 or CD 20 together with CD23 and CD5\r\n* Less than 55% atypical cells Documented expression of CD30 on tumor cells Patients must have histological confirmation of the diagnosis (it is recommended that the immunohistochemical panel includes: CD45, CD20, CD30, CD15, CD10, BCL6, BCL2, MUM-1), and in addition have a dominant mass within the anterior mediastinum. Previous infusion of CD19 CAR T cells at another institution CD30 positive tumor (can be pending at this time) Diagnosis - CD30+ HL or CD30+ NHL After Dose Escalation: any patient (children or adults) with relapsed CD30+ HL or CD30+ NHL or newly diagnosed patients unable to receive or complete standard therapy OR diagnosis of relapsed/refractory CD30+ HL or CD30+ NHL with a treatment plan that will include high dose therapy and autologous stem cell transplantation CD30 positive tumor Evidence of HCL by flow cytometry of blood or a solid (lymph node) mass, confirmed by the Laboratory of Pathology, National Cancer Institute (NCI), including positivity for CD19, CD22, CD20, and CD11c; patients with flow cytometry consistent with HCL variant (HCLv) are eligible, including those with CD25 and/or CD103 negative disease Prior treatment with the following agents: a) Tumor necrosis factor receptor (TNFR) agonists, including OX40, cluster of differentiation (CD)27, CD137 (4-1BB), CD357 (glucocorticoid-induced TNFR family-related gene) at any time. b) TLR4 agonist at any time. c) Anticancer therapy or investigational therapy within 30 days or 5 half-lives of the drug, whichever is shorter. d) Prior radiation therapy: permissible if at least 1 non-irradiated measurable lesion is available for assessment according to RECIST version 1.1 or if a solitary measurable lesion was irradiated, objective progression is documented. A wash out of at least 14 days before start of study treatment for radiation of any intended use to the extremities for bone metastases and 28 days for radiation to the chest, brain, or visceral organs is required. Any prior exposure to Hu5F9?G4 or other CD47 targeting agents. Patients must have a confirmed diagnosis of CLL as defined by the International Workshop on CLL 2008 (iwCLL2008) criteria below:\r\n* Presence of at least 5 x 10^9 B lymphocytes/L (5000/uL) in the peripheral blood\r\n* Morphologically, the lymphocytes must appear of small to moderate size with < 55% prolymphocytes, atypical lymphocytes or lymphoblasts\r\n* The clonality and immunophenotype of the circulating B-lymphocytes must be confirmed by flow cytometry to express cluster of differentiation (CD)5, CD23, CD19, CD20, CD52 and either kappa or lambda light chain Diagnosis of:\r\n* Biopsy-proven small lymphocytic lymphoma (SLL) , or\r\n* Diagnosis of chronic lymphocytic leukemia (CLL) with a clonal B-cell population in the peripheral blood with immunophenotyping consistent with CLL as follows:\r\n** The population of lymphocytes share both B-cell antigens (CD19, CD20 [typically dim expression], or CD23) as well as CD5 in the absence of other pan-T-cell markers (CD3, CD2, etc.)\r\n** Clonality as evidenced by kappa or lambda light chain expression (typically dim immunoglobulin expression) or other genetic method (e.g. IGHV analysis)\r\n** Before diagnosing CLL or SLL, mantle cell lymphoma must be excluded by demonstrating a negative fluorescence in situ hybridization (FISH) analysis for t(11;14)(IgH/CCND1) CD30-positivity by immunohistochemistry of >= 1% Confirmed diagnosis of MCL with CD20 and cyclin D1 positivity in tissue biopsy. Prior treatment with CD47 or signal regulatory protein alpha (SIRP?) targeting agents Patients must have newly diagnosed T-lymphoblastic leukemia (T-ALL) or T-lymphoblastic lymphoma (T-LLy) stages II-IV \r\n* Note: a diagnosis of T-ALL is established when leukemic blasts lack myeloperoxidase or evidence of B-lineage derivation (cluster of differentiation [CD]19/CD22/CD20), and express either surface or cytoplasmic CD3 or two or more of the antigens CD8, CD7, CD5, CD4, CD2 or CD1a, and are present either in peripheral blood or > 25% in the bone marrow; if surface CD3 is expressed on all leukemic cells, additional markers of immaturity, including terminal deoxynucleotidyl transferase (TdT), CD34 or CD99 will be assessed for expression; cases with uncertain expression will receive additional review within the appropriate Children's Oncology Group (COG) reference laboratory\r\n* For T-LLy patients with tissue available for flow cytometry, the criterion for diagnosis should be analogous to T-ALL; for tissue processed by other means (i.e. paraffin blocks), the methodology and criteria for immunophenotypic analysis to establish the diagnosis of T-LLy defined by the submitting institution will be accepted Patients must have histologically or flow cytometry confirmed diagnosis of B-cell chronic lymphocytic leukemia/small lymphocytic lymphoma (B-CLL/SLL) according to the IWCLL 2008 criteria. The malignant B cells must co-express CD5 with CD19 or CD20. Patients who lack CD23 expression on their leukemia cells should be examined for (and found NOT to have) either t(11;14) or cyclin D1 overexpression, to rule out mantle cell lymphoma. Severe CD as defined by one of the following:\r\n* CDAI >= 250\r\n* Need for total parenteral nutrition to maintain weight\r\n* Recurrent intestinal inflammation caused by CD following surgical resection Subjects must complete the filgrastim (G-CSF)/plerixafor mobilization of peripheral blood progenitor cells and must have collected at least 7.5 x 10^6 CD34+ cells/kg, sufficient for preparation of both, a back-up of 2.5 x 10^6 CD34+ cells and the research product Diagnosis of CD-30 positive GCT; CD30 expression will be tested by immunohistochemistry (IHC) in archival or fresh tumor tissue as is routinely done for diagnosis At least 10% of the cells obtained from lymph node, or extranodal sites must react with anti-CD25 (anti-Tac) on immunofluorescent or immunoperoxidase staining; patients with CD25-positive infiltrating T cells will be eligible even if their Hodgkin’s (Reed-Sternberg) cells are CD25-negative Autologous graft with a minimum of >= 3.0 x 10^6 CD34+ cells/kg; not CD34 selected Patients must have a histologically confirmed diagnosis of B-NHL, T-NHL, or HL; CD45 antigen expression must be documented on tumor specimens in all cases except HL, in whom histologic demonstration of CD45+ cells adjacent to the Reed Sternberg cells is required Histopathological evidence of CD25+ HCL confirmed by the National Institutes of Health (NIH) pathology department; this will require a monoclonal population of peripheral malignant lymphocytes that are CD25 positive by fluorescence activated cell sorting (FACS) with anti-CD25 antibody; positive expression in a FACS assay is defined as more than 2 times the mean fluorescence intensity (MFI) of the control antibody by FACS; HCLv (HCL variant) is usually CD25 negative, and eligibility would require CD25+ HCLv Prior treatment with CD19 directed agents unless CD19 expression is confirmed after completion of CD19-directed treatment The lymphoma must express the cluster of differentiation (CD)20 antigen by either flow cytometry using anti-CD20 antibodies or by immunoperoxidase staining of paraffin sections; a report providing confirmation of CD20 expression must be submitted Patients who have an option for any treatment with proven clinical benefit for CD25-positive AML or CD25-positive ALL at current state of disease. CD4+ T-cell count >= 100 cells/uL Expression of CD30 Histologically documented CD20-positive lymphoma as determined by the local laboratory Known CD20-negative status at relapse or progression Patients must have histologically confirmed malignancy that is metastatic or unresectable and for which standard curative or palliative measures do not exist or are no longer effective; previously treated, pathologically confirmed chronic lymphocytic leukemia (CLL) or small lymphocytic lymphoma (SLL) that requires as per the National Cancer Institute (NCI) Working Group Guidelines for the treatment of CLL; the diagnosis of CLL is defined by the presence of 5 x 10^9 clonal B-lymphocytes/L in the peripheral blood; clonality of the lymphocyte population is established with flow cytometry and the demonstration of the following surface markers: CD5, CD23, CD19, CD20 and the presence of either kappa or lambda immunoglobulins; the diagnosis of SLL may be made with the demonstration of < 5 x 10^9 clonal B-lymphocytes/L in the peripheral blood, with the clinical or radiographic features of enlarged lymph nodes or organomegaly, and the demonstration of SLL cells in the lymph node biopsy; institutional flow cytometry or immunohistochemistry must confirm CD20 antigen expression; patients may not have had a history of Richter’s transformation CD30 immunohistochemical staining using the anti-CD30 Becton Dickinson monoclonal (BerH2) antibody must be available on the most recent biopsy specimen; during dose escalation, patients can be either CD30 positive or CD30 negative; during dose expansion, 15 patients must be CD30 positive and 15 patients must be CD30 negative Immunotherapy within 6 months of C1D1 Available CD34+ stem cells Prior treatment with a therapeutic agent targeting CD19 and/or CD3 Patients must be diagnosed with CLL in accordance with International Workshop on Chronic Lymphocytic Leukemia (IWCLL) 2008 criteria that includes all of the following:\r\n* >= 5 x 10^9 B lymphocytes (5000/uL) in the peripheral blood\r\n* On morphologic review, the leukemic cells must be small mature lymphocytes, and prolymphocytes must not exceed 55% of the blood lymphocytes\r\n* CLL cells on immunophenotype (performed locally) must reveal a clonal B-cell population, which express the B cell surface markers of cluster of differentiation (CD)19 and CD20, as well as the T-cell antigen CD5; patients with bright surface immunoglobulin expression or lack of CD23 expression in > 10% of cells must lack t(11;14) translocation by interphase cytogenetics Diagnosis of recurrent CD30+ HL or CD30+ NHL, or newly diagnosed patients unable to receive or complete standard therapy OR diagnosis of relapsed/refractory CD30+ HL or CD30+ NHL with a treatment plan that will include high dose therapy and stem cell transplantation CD30 positive tumor (result can be pending at this time) Diagnosis - CD30+ HL or CD30+ NHL:\r\n* During the dose escalation phase: only adult patients with active disease failing standard therapy\r\n* After dose escalation: any patient (children or adults) newly diagnosed, unable to receive or complete standard therapy OR diagnosis of relapsed/refractory CD30+ HL or CD30+ NHL with a treatment plan that will include high dose therapy and autologous stem cell transplantation CD30 positive tumor Clinical and phenotypic verification of B cell chronic lymphocytic leukemia (CLL) and measurable disease; immunophenotyping of the leukemic cells (blood or marrow) must demonstrate a monoclonal (or light chain positive) B cell population with immunophenotype consistent with CLL (e.g., co-expressing CD19 and CD5) Known CD20-negative status at relapse or progression; CNS lymphoma or leptomeningeal infiltration Uncontrolled angina within 3 months before C1D1. Diagnosis of CLL including:\r\n* Monoclonal B-cells co-expressing >= one B-cell marker (cluster of differentiation [CD]19, CD20, or CD23) and CD5 in peripheral blood or lymph node Prior treatment with TNFRSF agonists including OX40, CD27, CD137 (4-1BB), CD357 (GITR) . Subjects must have CLL/SLL, as documented by a history at some point in time of an absolute peripheral blood B cell count > 5000/ul, with a monoclonal B cell population co-expressing cluster of differentiation (CD)19, CD5, and CD23, or if CD23 negative, then documentation of the absence of t(11;14) or cyclin D1 overexpression; alternatively patients with lymphadenopathy in the absence of circulating disease will also be eligible for this study if lymph node biopsy or bone marrow biopsy establishes the diagnosis of CLL with the above immunophenotype Immune compromised patients including but not limited to: systemic immune suppressive medications within 6 weeks of enrolling; HIV-positive and below normal CD4 lymphocytes (less than 500 cells per microliter). Patients must be tested for HIV seropositivity and CD4 lymphocyte count to be eligible for the study Diagnosis of CLL according to the National Cancer Institute (NCI)/Internal Workshop on Chronic Lymphocytic Leukemia (IWCLL) criteria or small lymphocytic lymphoma (SLL) according to the World Health Organization (WHO) criteria; this includes previous documentation of:\r\n* Biopsy-proven small lymphocytic lymphoma or\r\n* Diagnosis of CLL according to the NCI/IWCLL criteria as evidenced by all of the following:\r\n** Peripheral blood lymphocyte count of greater than 5 x 10^9/L\r\n** Immunophenotype consistent with CLL defined as:\r\n*** The predominant population of lymphocytes share both B-cell antigens (cluster of differentiation [CD]19, CD20 [typically dim expression], or CD23) as well as CD5 in the absence of other pan-T-cell markers (CD3, CD2, etc)\r\n*** Clonality as evidenced by kappa or lambda light chain restriction (typically dim immunoglobulin expression)\r\n* Negative FISH analysis for t(11;14)(immunoglobulin heavy locus [IgH]/cyclin D1 [CCND1]) on peripheral blood or tissue biopsy (e.g. marrow aspirate) or negative immunohistochemical stains for cyclin D1 staining on involved tissue biopsy (e.g. marrow aspirate or lymph node biopsy) More than 2 x 10^6 autologous CD34+ cells/kg cryopreserved. The graft may not be CD34+ selected or otherwise manipulated to remove tumor or other cells. The graft can be collected at the transplanting institution or by a referring center SLL (lymphadenopathy and/or splenomegaly and < 5×10^9 CD19+ CD5+ clonal B lymphocytes/L [< 5000/µL] in the peripheral blood at diagnosis with measurable disease that is biopsy-proven SLL) If prior CD19-targeted therapy has been administered, subject must have CD19-positive disease confirmed by immunohistochemistry or flow cytometry since completing the prior CD19-targeted therapy. CD30 negative HL Current diagnosis of primary cutaneous CD30-positive T-cell lymphoproliferative disorders and lymphomas or mycosis fungoides Investigational or biologic therapies within 3 weeks of C1D1 Tumors must be CD20+ on prior pathologic analysis Tumor cell negative for CD20 Patients must have histologically or cytologically confirmed cluster of differentiation (CD)5 positive (+)/CD20+ B-cell chronic lymphocytic leukemia or small lymphocytic lymphoma; the diagnosis of CLL is based upon the National Comprehensive Cancer Network (NCCN) guidelines; any outside pathology slides used as inclusion criteria for the patient will be reviewed at this institution to confirm the diagnosis; the patient must meet all of the following CLL criteria to participate in this study: \r\n* Absolute lymphocyte count > 5000/uL \r\n* CD20+ and CD5+ \r\n* Bone marrow lymphocytes >= 30% \r\n* Or previous confirmed diagnosis of CLL/SLL with less than 5000/ul or less than 30% lymphocytes in bone marrow (BM) Histologically confirmed diagnosis of mantle cell Non-Hodgkin's Lymphoma with cyclin D1 overexpression by immunohistochemistry, and a characteristic immunophenotypic profile with CD5(+), CD23(-), CD20(+), and CD10(-). In tumor tissues with negative cyclin D1, evidence of cyclin D2 or D3 overexpression by immunohistochemistry will be acceptable. Patients must have a biopsy confirmed diagnosis based on a combination of histological and clinical criteria of CD30+ lymphomatoid papulosis, CD30+ primary cutaneous anaplastic large T-cell lymphoma (pc-ALCL), or CD30+ mycosis fungoides for the phase II trial; there is no specific limit or validated amount other than positive cells on immunohistochemistry (IHC) cells in tumor cells Diagnosis of chronic lymphocytic leukemia:\r\n* Lymphocytosis of >= 5,000 cells/ul; AND\r\n* Marrow aspirate with >= 30% of mononuclear cells being lymphocytes; AND\r\n* Flow cytometry demonstrating monoclonal B-cells co-expressing >= one B cell marker (cluster of differentiation [CD]19, CD20, or CD23) and CD5 Previous therapy directed against CD19, such as MAbs or MAb conjugates Patients with less than 50% donor CD3 peripheral blood chimerism on two separate, consecutive evaluations; the two evaluations must be at least 14 days apart OR patients with absolute decreases of donor CD3 peripheral blood chimerism of >= 20% if the second test shows < 50% donor CD3 cells; the two evaluations must be at least 14 days apart Patients must have persistent donor CD3 cells (>= 5% donor CD3 cells by a deoxyribonucleic acid [DNA]-based assay that compares the profile of amplified fragment length polymorphisms [ampFLP] [or fluorescent in situ hybridization (FISH) studies or variable number of tandem repeats (VNTR)]) Chemotherapy within 28 days prior to C1D1 CD123-detectable leukemia Participants must have available archival or fresh tumor tissue or both to submit to a central laboratory for CD38 assay. Expression of CD38 is measured by immunohistochemistry on fresh or archived tumor sample by central assessment using a CD38 investigational IHC assay under development: a) Stage 1: participants whose tumors are more than or equal to (>=) 50 percent (%) positive for CD38, b) Stage 2: participant has less than (<) 50% CD38+ or greater than (>) 50% CD38+ depending on the distribution of CD 38 expression of enrolled participants during Stage 2. The sponsor will advise on which eligibility criterion is permitted during the enrollment period Evidence of CD19 expression LVEF < 40% as determined by echocardiogram performed at screening or within 90 days prior to C1D1 Any level of CD56 expression will be considered sufficient for enrollment on this study Histologically confirmed diagnosis of B-lineage ALL. Verification of CD22 expression is not required Diagnosis of CD20-positive FL: CD20 immunophenotyping performed ?2 years prior to randomization Unable to generate antigen-specific MCPyV TAg-specific CD8+ T cells for infusions Primary cutaneous CD30+ disorders: ALCL and lymphomatoid papulosis Known CD20-negative status at relapse or progression Diagnosis of B-cell CLL/SLL including:\r\n* Lymphocytosis of monoclonal B-cells co-expressing >= one B-cell marker (CD19, CD20, or CD23) and CD5 in peripheral blood or lymph node AND\r\n* Bone marrow with >= 30% mononuclear cells having the CLL/SLL phenotype CD19 CAR-T cells based therapies Diagnosis of:\r\na) CLL according to the National Cancer Institute (NCI) criteria\r\nb) Small lymphocytic lymphoma (SLL) according to the World Health Organization (WHO) criteria\r\nc) MBL according to the consensus criteria\r\n* This includes previous documentation of:\r\n** Biopsy-proven small lymphocytic lymphoma or\r\n** Diagnosis of CLL or MBL as evidenced by all of the following:\r\n*** Clonal B-cell population in the peripheral blood with immunophenotyping consistent with CLL defined as:\r\n**** The population of lymphocytes share both B-cell antigens (cluster of differentiation [CD]19, CD20 [typically dim expression], or CD23) as well as CD5 in the absence of other pan-T-cell markers (CD3, CD2, etc)\r\n**** Clonality as evidenced by k (kappa) or lambda light chain expression (typically dim immunoglobulin expression) or other genetic method (e.g. immunoglobulin heavy chain variable [IGHV] analysis)\r\nNOTE: splenomegaly, hepatomegaly, or lymphadenopathy are not required for the diagnosis of CLL \r\n*** Patients with a peripheral blood B-cell lymphocyte count of < 5 x 10^9/L and no evidence of lymphadenopathy or organomegaly will be classified as MBL; patients with a peripheral blood B-cell lymphocyte count of < 5 x 10^9/L who have evidence of lymphadenopathy will be classified as SLL; patients with a peripheral blood B-cell lymphocyte count >= 5 x 10^9/L will be considered to have CLL\r\n*** Before diagnosing MBL, CLL or SLL, mantle cell lymphoma must be excluded by demonstrating a negative fluorescence in situ hybridization (FISH) analysis for t(11;14)(IgH/cyclin D 1 [CCND1]) on peripheral blood or tissue biopsy or negative immunohistochemical stains for cyclin D1 on involved tissue biopsy Confirmed positive CD70 expression on tumor tissue Washout from any prior investigational therapy of at least five times the T1/2 prior to C1D1 Washout of at least 2 weeks from the most recent radiation treatment prior to C1D1 Is within 2 months of transplant from C1D1 CD22 expression A diagnosis of CLL defined by a circulating B-lymphocyte count of greater than or equal to 5,000/uL at study entry or at any time in the past and flow cytometry confirmation of immunophenotype with CD5, CD19, CD20, CD23, CD79b, and surface Ig prior to first dose of study treatment. Have FSH levels indicative of postmenopausal state (i.e., 30-120 IU/L) documented within 21 days of C1D1. Recurrent disease:\r\n* Patient > 5 years old (yo) must have recovered CD4 count to > 350 cells/mm3 OR have disease free interval > one year from completion of cytotoxic therapy\r\n* Patients < 5yo must have recovered CD4 count to > 360 cells/mm3 OR have disease free interval > six months from completion of cytotoxic therapy Multiple recurrences are allowable as long as CD4 count or disease-free intervals have been met Histologically-confirmed CD30-positive disease; tissue from the most recent post diagnostic biopsy of relapsed/refractory disease must be available for confirmation of CD30 expression via slides or tumor block. Predominant donor chimerisms as measured by CD3 and CD33 (or other myeloid marker) Alemtuzumab or any equivalent in vivo T-cell depleting agent (or CD34+ selection) Greater than 25% of blasts must be CD33 positive. Greater than 25% of blasts must be CD33 positive. Prior to Segment B enrollment, participants on combination anti-retroviral therapy (cART) will be required to have a minimum CD4 count of >= 200 and participants not on cART will be required to have a minimum CD4 count of >= 350 to be eligible for the study; participants not currently on cART who have a CD4 count >= 200 and who agree to start cART immediately will be eligible for participation; laboratory data should be obtained within 16 weeks prior to Segment B enrollment Subjects must have CD34+ collection which allows reinfusion of ?1.5 x 106 and ?5.0 x 106 CD34+ cells/kg Possession of a CD/DVD player or ability to play a CD Recipients of CD34 selected grafts or other manipulated grafts (with any form of ex vivo T cell depletion) will be eligible to enroll if they have a CD3 count > 100; (please note: post-transplant Cytoxan for haploidentical transplants is allowable) FOR THE 31 SUBJECTS ENROLLED IN YEAR 1: Recipients of CD34 selected grafts or other manipulated grafts (with any form of ex vivo T cell depletion) will be eligible to enroll if they have a CD3 count > 100; (please note: post-transplant cytoxan for haploidentical transplants is allowable) CD4 count < 200 cells/mm^3 within 6 months of entry\r\n* Note: This refers to any CD4 obtained for routine care; documentation of CD4 is not required prior to entry CD30+ disease (result can be pending at the time of cell procurement, but must be confirmed prior to treatment with ATLCAR.CD30 cells); NOTE: CD30 + disease requires documented CD30 expression by immunohistochemistry based on the institutional hematopathology standard Required prior to infusion of ATLCAR.CD30 cells: AST =< 3 times ULN Required prior to infusion of ATLCAR.CD30 cells: Serum creatinine =< 1.5 times ULN Study 2 - CLL/SLL\r\n* Newly diagnosed (< 12 months from pre-registration on this study) CLL according to the National Cancer Institute (NCI) criteria or SLL according to the World Health Organization (WHO) criteria; this includes previous documentation of:\r\n** Biopsy-proven small lymphocytic lymphoma\r\n** Diagnosis of CLL according to NCI working group criteria as evidenced by all of the following:\r\n*** Peripheral blood lymphocyte count of > 5,000/mm^3; if present, prolymphocytes should be < 55%\r\n*** Immunophenotyping consistent with CLL defined as:\r\n**** The predominant population of lymphocytes share both B-cell antigens (cluster of differentiation [CD]19, CD20, or CD23) as well as CD5 in the absence of other pan-T-cell markers (CD3, CD2, etc.)\r\n**** Dim surface immunoglobulin expression\r\n**** Restricted surface kappa or lambda light chain expression\r\n*** Before diagnosing CLL or SLL, mantle cell lymphoma must be excluded by demonstrating a negative fluorescent in situ hybridization (FISH) analysis for t(11;14)(immunoglobulin H [IgH]/cyclin D 1 [CCND1]) on peripheral blood or tissue biopsy or negative immunohistochemical stains for cyclin D1 on involved tissue biopsy\r\n* Rai stage 0 or 1\r\n* Previously untreated\r\n* Asymptomatic with the plan for observation\r\n* Life expectancy of at least 24 months\r\n* Willing to provide tissue for correlative research purposes Patients must have a plan to receive a CD34-selected peripheral blood stem cell graft Prior treatment with CD47 or signal regulatory protein alpha (SIRP?) targeting agents (with exception of Hu5F9-G4 for patients in the Rollover cohort). Phenotypic studies (obtained within 8 weeks prior to study drug administration) from peripheral blood showing CD3+, CD57+ cells >400/mm³ or CD8+ cells >650/mm³. Pathologically confirmed MCL (in tumor tissue), with documentation of either overexpression of cyclin D1 in association with other relevant markers (eg, CD19, CD20, PAX5, CD5) or evidence of t(11;14) as assessed by cytogenetics, fluorescent in situ hybridization (FISH), or polymerase chain reaction (PCR) Radiation, chemotherapy, or immunotherapy or any other anticancer therapy (including investigational therapies) ? 2 weeks prior to C1D1. Localized radiation to a single site at least 1 week before C1D1 is permitted. Glucocorticoids within 2 weeks of C1D1 are permitted. Patients on long-term glucocorticoids during Screening do not require a washout period but must be able to tolerate the specified dexamethasone dose in this study. All patients who have received anti-CD19 directed CART therapy and completed or discontinued early from a Novartis sponsored treatment protocol that utilized CD19-directed CART cells or from any CD19 CART trial sponsored by the University of Pennsylvania with which Novartis has a contractual agreement to co-develop the CAR technology. Patients must have histologically or cytologically confirmed by the local institution CD19+ precursor B-acute lymphoblastic leukemia (pre-B cell ALL) OR CD19+ mixed phenotype acute leukemia (MPAL): a) with relapse following or refractory to at least one prior line of therapy if older than 21 years; b) in second or higher relapse or refractory to at least two prior lines of therapy if 21 years old and younger (16-21); c) or they must have a new diagnosis of pre-B cell ALL or CD19+ MPAL but are >= 60 years old and are either not a candidate for or do not wish to receive traditional induction chemotherapy Patients who were treated with blinatumomab in the past will be allowed on the study as long as they have persistent CD19 expression on leukemia cells and did not experience unacceptable toxicities with prior blinatumomab administration; patients who were treated with chimeric antigen receptor (CAR)-modified T cells targeting CD19 in the past will be allowed on the study as long as they have persistent CD19 expression on leukemia cells Clinical and phenotypic verification of B cell CLL or small lymphocytic lymphoma (SLL) and measurable disease; immunophenotyping of the leukemic cells (blood, lymph node, or bone marrow) must demonstrate a monoclonal (or light chain positive) B cell population with immunophenotype consistent with CLL (e.g.: co-expressing CD19, CD5, and CD23)