Patient must be deemed eligible to proceed with high-dose chemotherapy and autologous stem cell transplantation by local transplant center At least 6 weeks from autologous stem cell transplantation Must be ineligible for high dose therapy/ stem cell transplantation Participants may not be receiving any other non-Food and Drug Administration (FDA) approved study agents at the start of conditioning for stem cell transplantation; patients may receive non-FDA approved agents at the time of screening/enrollment as long as such agent(s) will be discontinued by the start of conditioning for transplantation Eligible for Autologous Hematopoietic stem cell transplantation according at the investigator discretion. B-cell lymphoma classified as either of the following: R/R FL after treatment with at least one prior chemoimmunotherapy regimen that included an anti-cluster of differentiation 20 (CD20) monoclonal antibody; R/R DLBCL after treatment with at least one prior chemoimmunotherapy regimen that included an anti-CD20 monoclonal antibody in participants who are not eligible for second line combination chemotherapy and autologous stem-cell transplantation, have failed second line combination chemotherapy, or experienced disease progression following autologous stem-cell transplantation Patients who completed single autologous stem cell transplant after completion of at most 2 induction regimens (excluding dexamethasone alone) and are in at least stable disease compared to pre-induction in the first 100 days after stem cell transplantation For rituximab in combination with polatuzumab vedotin and lenalidomide (R + Pola + Len) treatment group: R/R DLBCL after treatment with at least one prior chemoimmunotherapy regimen that included an anti-CD20 monoclonal antibody in patients who are not eligible for autologous stem-cell transplantation or who have experienced disease progression following treatment with high-dose chemotherapy plus autologous stem-cell transplantation Previous hematopoietic stem cell transplantation; patients can have had prior relapsed disease as long as they have never been previously transplanted Institutional criteria for and have institutional approval to undergo autologous peripheral blood stem cell transplantation NOTE: There is no limit to the prior number of chemotherapy regimens; patients with prior autologous or allogeneic stem cell transplantation, as well as prior therapy with cyclophosphamide or alemtuzumab, are eligible Patients who have ever received an Autologous Hematopoietic Stem Cell Transplantation (autoHSCT) ARE eligible. Patients who are scheduled to receive an Autologous Hematopoietic Stem Cell Transplantation (autoHSCT) are NOT eligible Recipients of prior autologous hematopoietic stem cell transplantation are ineligible if disease recurrence occurred less than 6 months from their autologous stem cell transplant. Hematological malignancy has been previously treated, has relapsed after or progressed during prior therapy, and has limited potential for benefit from currently available therapy including hematopoietic stem cell transplantation. Other high-risk hematologic malignancies eligible for stem cell transplantation per institutional standard One or more prior lines of chemoimmunotherapy and/or monotherapy with rituximab or other anti-cluster of differentiation (CD)20 antibody; patients may have had a prior autologous stem cell transplant but not prior allogeneic stem cell transplantation Patient has an appropriate donor identified for hematopoietic stem cell transplantation Deemed eligible for autologous stem cell transplantation (ASCT) by standard institutional criteria For ALL: patients must belong to one of the following ‘high risk’ categories: \r\n* Primary refractory as defined by failure to achieve CR after induction and at least one salvage therapy \r\n* Second or subsequent relapse \r\n* Relapse after allogeneic or autologous stem cell transplantation (requirement for second relapse does not apply post-transplant) \r\n* All variants of ALL including T-ALL, B / myeloid, lymphoblastic leukemia lymphoma are eligible Cohort #1: considered eligible for high-dose chemotherapy followed by autologous stem cell transplantation (ASCT) High-dose chemotherapy followed by stem cell transplantation (autologous or allogeneic). Patients with histologically-confirmed symptomatic multiple myeloma or AL amyloidosis undergoing autologous hematopoietic cell transplantation (HCT) with melphalan 140 or 200 mg/m^2 Patients with prior salvage chemo-immunotherapy, radiation therapy, autologous transplantation are included High dose chemotherapy followed by autologous stem cell transplantation within 90 days prior to initiating study treatment; Patients who are candidate for an autologous or allogeneic stem cell transplantation (SCT) will be allowed to receive the study drugs as a “bridge” to transplantation if candidates for transplant Patients with HL or DLBCL must refuse or not be candidates for curative autologous stem cell transplantation Received prior autologous stem cell transplantation as first line therapy for multiple myeloma with subsequent disease relapse/progression Refractory to or intolerant of lenalidomide maintenance following first autologous stem cell transplantation; refractory is defined as disease relapse/progression on therapy or within 60 days of completing therapy; intolerance is defined as the inability to administer >= 10 mg per day due to toxicity Candidate for second autologous stem cell transplantation per local institution’s guidelines with at least 2 x 10^6/kg CD34+ autologous stem cells available for transplantation Prior treatment with chimeric antigen receptor (CAR) T cells or other forms of adoptive cellular therapy, with the exception of autologous stem cell transplantation History of prior autologous hematopoietic cell transplantation Hematological malignancy has been previously treated, has relapsed after or progressed during prior therapy, and has limited potential for benefit from currently available therapy, including hematopoietic stem cell transplantation. Patient participants who have undergone autologous stem cell transplantation (autoSCT) are eligible provided that they are >= 4 weeks from stem cell infusion Patients are planned for treatment with high dose melphalan and autologous hematopoietic cell transplantation (HCT) Autologous stem cell transplantation within 12 weeks prior to study entry RESEARCH SAMPLE COLLECTION: Considered for high-dose melphalan followed by autologous hematopoietic cell transplantation and undergoing pre-HCT workup Diagnosis of multiple myeloma undergoing planned autologous stem cell transplantation Patients must have undergone hematopoietic stem cell transplantation and have moderate to severe chronic GVHD as defined by the National Institute of Health (NIH) consensus criteria Not suitable for, or declined high dose chemotherapy and autologous stem cell transplantation (ASCT). Prior autologous stem cell transplantation. Patients who have received prior stem cell transplantation will be allowed to enroll as long as prior transplantation has been at least 3 months before enrollment in the trial and any transplant related toxicities have subsided to grade 1 or less Patients must have failed at least 1 prior regimen before ibrutinib (not including single agent rituximab or single agent corticosteroids)\r\n* Note: any relapse after prior autologous stem cell transplantation (SCT) will make the patient eligible regardless of other prior therapy Patients who have undergone autologous stem cell transplantation within 3 months from study entry Tandem autologous transplantation DONOR SELECTION: Not applicable; this protocol employs autologous transplantation, utilizing the patient’s own hematopoietic stem cells obtained from either the peripheral blood or bone marrow Patients with B-lineage ALL at least marrow CR in Salvage 1 and beyond with molecular failure at any time point after 1 month of salvage therapy are allowed, including patients who received prior allogeneic stem cell transplantation Patients who have poor or no graft function post stem cell transplantation Patients must have undergone autologous stem cell transplantation, within 18 months of initiation of induction therapy for newly diagnosed myeloma Prior autologous hematopoietic progenitor cell transplantation if the conditioning regimen included total body irradiation Patients should have received single autologous stem cell transplantation 60-120 days prior to enrollment to the trial DIAGNOSIS REQUIREMENT FOR PHASE I PATIENTS: High-risk AML (by European Leukemia Net [ELN] criteria) in complete remission (CR) and has either refused hematopoietic stem cell transplantation OR is currently not eligible for hematopoietic stem cell transplantation OR for whom hematopoietic stem cell transplantation is being reserved for later relapse; this is inclusive of patients with minimal residual disease evidenced by cytogenetics, molecular testing, and/or flow cytometry OR Patients must be eligible to undergo autologous stem cell transplantation by standard institutional criteria Patients must have relapsed after high-dose therapy and autologous transplantation or be ineligible for high-dose therapy and autologous transplantation; patients that have failed autologous transplantation are those with persistent disease > 30 days after transplant; those ineligible for autologous transplant include those with chemoresistant disease (i.e., patients who have not achieved a partial response or better with their most recent chemotherapy regimen), are expected to have a poor outcome from autologous transplant (e.g., DLBCL relapsing within one year of rituximab, cyclophosphamide, doxorubicin hydrochloride, vincristine sulfate, prednisone [R-CHOP]-like chemotherapy, double hit lymphoma, v-myc myelocytomatosis viral oncogene homolog (avian) positive [MYC+] lymphoma, persistent positron emission tomography [PET] positivity after chemotherapy), are unable to collect sufficient or tumor-free autologous stem cells per Seattle Cancer Care Alliance (SCCA) standard practice, are unable to tolerate the high-dose autologous conditioning regimens, or who refuse a high-dose autologous transplant regimen Multiple myeloma with disease in the following categories:\r\n* Patients with relapsed multiple myeloma following autologous stem cell transplantation who have achieved at least partial response following additional chemotherapy\r\n* Patients with high risk cytogenetics at diagnosis must have achieved at least a partial response following autologous stem cell transplantation; patients must have complex karyotype, del17p, t4;14 and/or t14;16 by fluorescent in situ hybridization (FISH) and/or del13 by karyotyping Multiple myeloma (MM) stage II or III patients who have progressed after an initial response to chemotherapy or autologous hematopoietic stem cell transplantation (HSCT) or MM patients with refractory disease who may benefit from tandem autologous-nonmyeloablative allogeneic transplant Patients with stored autologous stem cells will be allowed Minimum donor and stem cell requirements for high-risk patients undergoing stem cell transplantation: Patients must be ineligible for autologous transplantation due to prior autologous transplant, an inadequate autologous stem cell harvest, inability to withstand a myeloablative preparative regimen, or clinically aggressive/high risk disease Stem cells: patients must have an autologous hematopoietic stem cell product cryopreserved and available for re-infusion after MIBG treatment; the minimum dose for peripheral blood stem cells is 2 x 10^6 CD34+ cells/kg Autologous stem cell transplantation less than 90 days prior to study day 1 Patients who have received prior stem cell transplantation will be allowed to enroll as long as prior transplantation has been at least 3 months before enrollment in the trial and any transplant related toxicities have subsided to grade 1 or less Have received or are ineligible for immediate established curative regimens, including stem cell transplantation Patients who have undergone autologous stem cell transplantation (ASCT) are eligible provided that they are >= 4 weeks from stem cell infusion and meet other eligibility criteria Eligible for autologous transplantation Prior high-dose chemotherapy and autologous hematopoietic cell transplantation (HCT)(s) is (are) allowed Considered by the investigator to be eligible for high-dose chemotherapy (HDT) and autologous stem cell transplantation (ASCT) according to the institution's criteria based on age, medical history, cardiac and pulmonary status, overall health and condition, co-morbid condition(s), physical examination, and laboratory studies Completion of, if applicable, an autologous stem cell transplantation (ASCT) at least 3 months prior to first dose of study drug. Not a candidate for curative therapy or hematopoietic stem-cell transplantation (either due to disease burden, fitness, or preference). Has received autologous stem cell transplantation (ASCT) within 12 weeks before the date of randomization. Participant must be ineligible or unwilling to undergo stem cell transplantation at time of study entry Patients who have undergone autologous hematopoietic stem cell transplantation are eligible once they have recovered from all toxicities from therapy Subject must have documented diagnosis with previously untreated (for cohort C, the induction and consolidation treatment along with the first Autologous stem cell transplantation (ASCT) are allowed), symptomatic Multiple Myeloma (MM) as defined by the criteria below: Autologous hematopoietic cell infusion after high dose therapy At least 60 days Patients must have undergone autologous hematopoietic stem cell transplantation (AHCT) and achieved engraftment by day (D)60-180 as evidenced by absolute neutrophil count (ANC) > 1000/mcL and platelets (Plt) > 75,000/mcL Eligible for autologous stem cell transplantation The subject must not be a candidate for potentially curative therapy including hematopoietic stem cell transplantation, except where one of the standard therapy regimen combinations may be used prior to transplantation per standard medical practice The disease needs to be in one of the following stages:\r\n* At diagnosis or in first relapse AND the patient is unable to receive conventional chemotherapy for his/her condition\r\n* In second or subsequent relapse\r\n* With residual disease after autologous, syngeneic or allogeneic hematopoietic stem cell transplantation (HSCT) Any previous autologous hematopoietic stem cell transplantation (HSCT) must have occurred at least 3 months prior to start of conditioning Diagnosed with AML and eligible for standard induction chemotherapy or stem cell transplantation For rituximab + Atezo + Pola treatment group: relapsed or refractory DLBCL after treatment with at least one prior chemoimmunotherapy regimen that included an anti-CD20 monoclonal antibody, in participants who are not eligible for second line combination (immuno-) chemotherapy and autologous stem-cell transplantation or who have failed second line combination (immuno-) chemotherapy or experienced disease progression following autologous stem-cell transplantation Eligible for autologous stem cell transplantation Has received high dose chemotherapy followed by autologous stem cell transplantation less than 90 days prior to first dose of study treatment Participants who are newly diagnosed and not considered candidate for high-dose chemotherapy with stem cell transplantation (SCT) due to: being age >=65 years, or in participants <65 years: presence of important comorbid conditions likely to have a negative impact on tolerability of high dose chemotherapy with stem cell transplantation Patients must have received at least two prior lines of therapy and also must be refractory to lenalidomide (defined as progression while on active therapy or within 60 days of discontinuation); one prior line of therapy may consist of all predetermined components of induction followed by autologous stem cell transplantation and maintenance Research participant has an indication to be considered for autologous stem cell transplantation Research participant did not have evidence of disease progression after salvage therapy and therefore underwent an autologous myeloablative transplantation with hematopoietic progenitor cell autologous (HPC[A]) rescue procedure Prior solid organ transplantation or allogenic stem cell transplantation (ASCT). However, previous autologous BM transplant (ABMT) or autologous peripheral blood stem cell transplant (PBSCT) is permitted. Patients for whom lenalidomide and dexamethasone treatment is appropriate and who are not eligible for high-dose therapy followed by stem-cell transplantation (HDT-SCT) for 1 or more of the following reasons: Patient is considered for autologous stem cell transplantation with full dose melphalan (200 mg/m^2) Patients with relapsed multiple myeloma following autologous stem cell transplantation who achieved < partial response following additional chemotherapy or who achieved < partial response (PR) at 3 months following autologous stem cell transplantation and patients with plasma cell leukemia at diagnosis Patient must have undergone autologous stem cell transplantation, with melphalan as a preparative regimen, within 12 months of initiation of induction therapy for newly diagnosed myeloma Patients who have previously received an autologous stem cell transplantation must be at least 4 weeks post-transplant before study drug administration and must have exhibited a full haematological recovery Patients who are eligible for autologous transplantation Patients with histologically confirmed CD20 positive B-cell non-Hodgkin lymphoma (NHL) who are candidates for autologous stem cell transplantation (SCT) Participants must have a diagnosis of multiple myeloma documented by having > 15% plasma cells on bone marrow biopsy and/or monoclonal protein in blood and/or urine; all patients must have disease which is either stable or responsive after a minimum of 2 cycles of conventional chemotherapy and slated to undergo autologous peripheral blood stem cell transplantation incorporating mobilization chemotherapy for peripheral blood stem cell collection Phase I: Patients with diagnosis of multiple myeloma at any stage of disease undergoing high dose chemotherapy and stem cell transplantation; Phase II: Patients with myeloma undergoing a first high dose chemotherapy and stem cell transplantation after achieving at least stable disease following induction therapy; any induction regimen prior to transplantation is allowed; no more than 2 prior lines of therapy prior to transplantation are allowed Patients with relapsed multiple myeloma following autologous stem cell transplantation must have achieved at least partial response following additional chemotherapy (cohort 1):\r\n* Patients are eligible if relapse occurs with complex/high-risk cytogenetics or occurs with normal cytogenetics but within 15 months following the autologous transplant Patients with high risk cytogenetics at diagnosis must have achieved at least very good partial response following autologous stem cell transplantation (cohort 2):\r\n* Patients must have complex karyotype, 1q25, del17p, t4;14 and/or t14;16 by fluorescence in situ hybridization (FISH) and/or del13 by karyotyping Patients achieving < partial response following preceding chemotherapy (cohort 1) or < very good partial response following autologous stem cell transplantation (cohort 2) Eligibility for potentially-curative therapy including hematopoietic stem-cell transplantation Progressive disease within 8 weeks of prior therapy or within 12 weeks after prior autologous stem cell transplantation Greater than approximately 6 months since autologous stem cell transplantation Subjects with AML in their first relapse following a remission >12 months in duration who are eligible for standard therapies (e.g. chemotherapy or stem cell transplantation); Patients with neutropenic fever who have existing malignancy or have undergone hematopoietic stem cell transplantation Patients who are candidates for autologous stem-cell transplantation due to primary refractory or first relapse of disease Patients with adequate autologous stem cell collection for transplantation (target >= 2.5 x 10^6 CD34+ cells/kg) Patients with newly diagnosed double hit in first complete remission, anytime during the first 3 months after chemoimmunotherapy followed by autologous stem cell transplantation if there was no evidence of progression Must have received at least 2 prior therapies, including a CD20 targeted therapy, alkylating agent or corticosteroid; subjects who are not eligible for high-dose chemotherapy and autologous stem cell transplantation (HD-ASCT) are eligible with exposure to at least 1 prior therapy. Waldstrom's Macroglobulinemia Dose Expansion Arm: “Relapsed or refractory” refers to patients who have received at least 1 prior treatment regimen for lymphoma (which may include prior autologous stem cell transplantation) and have demonstrated evidence of progressive disease by clinical and/or radiographic characteristics Eligible for autologous transplantation Previous autologous stem cell collection Eligible for autologous stem cell transplantation Prior therapy with severely myelotoxic regimens, including autologous and allogenic stem cell transplantation At least 1 prior specific therapeutic regimen, one of which should have included rituximab (patients previously eligible for transplantation: the salvage treatment followed by intensification and Autologous Stem Cell Transplant (ASCT) will be considered one regimen). Patients must have relapsed or progressed after at least one prior cytotoxic chemotherapy\r\n* Previous autologous or allogeneic stem cell transplantation is permitted\r\n* Previous treatment with either single agent panobinostat or lenalidomide is permitted Patients who are candidates for high dose chemotherapy and autologous stem cell transplantation with curative intent should not be enrolled Previous autologous stem cell transplantation within 6 months prior to randomization. The patient has received, or is receiving induction chemotherapy followed by Stem Cell Transplantation. The patient declines to undergo stem cell transplantation or Stem cell transplantation is not available to the patient due to cost or other reasons At the discretion of the principal investigator if he/she feels that the patient is unable to safely complete the study; specifically, patients must be considered medically eligible to undergo high dose chemotherapy and autologous stem cell transplantation Prior autologous stem cell transplantation < 1 month or allogenic stem cell transplantation < 3 months prior to C1D1. Prior allogeneic hematopoietic stem-cell transplantation if evidence of donor chimerism persists; patients with exclusively autologous hematopoiesis are eligible Patients who will undergo their first autologous hematopoietic stem cell transplantation (HSCT) procedure as treatment for multiple myeloma All patients must have a histologic or cytological diagnosis of ALL treated with stem cell transplantation; there are no restrictions on prior therapy Scheduled to receive conditioning chemotherapy followed by upfront or salvage autologous peripheral blood hematopoietic stem cell transplantation Patients with pathologically diagnosed who have received induction chemotherapy, with or without autologous stem cell transplantation (AuSCT), and who have qualified to receive lenalidomide-based maintenance therapy for their multiple myeloma (MM) Allogeneic stem cell transplantation using in vivo or ex vivo T cell depletion, either by cell manipulation or with T cell depleting antibodies (any anti-thymocyte globulin preparation or alemtuzumab given within 30 days of transplantation) Subjects with hematologic malignancies or recipients of a first allogeneic or autologous hematopoietic stem cell transplantation and presently clinically stable Eligible for haploidentical stem cell transplantation according to the investigator Autologous stem cell transplantation less than 90 days prior to study day 1 Patients receiving autologous stem cell transplantation must wait 8 weeks before initiation of study drug administration. Not a candidate for or refusing treatment with hematopoietic stem cell transplantation Not a candidate for, or refusing treatment with hematopoietic stem cell transplantation