Patients must be considered to be suitable RIC alloHCT candidates at the time of enrollment based on medical history, physical examination, and available laboratory tests. Specific testing for organ function is not required for eligibility but, if available, these tests should be used to judge eligibility. ELIGIBILITY CRITERIA FOR SCREENING AND MOLECULAR PROFILING (STEP ) REGISTRATION ELIGIBILITY CRITERIA REGISTRATION ELIGIBILITY CRITERIA: ELIGIBILITY CRITERIA FOR STEP ELIGIBILITY CRITERIA FOR STEP INDUCTION ELIGIBILITY CRITERIA-STEP POST-INDUCTION THERAPY ELIGIBILITY CRITERIA (PRIOR TO INTENSIFICATION-STEP ) ELIGIBILITY FOR CHEMOTHERAPY COHORT: ELIGIBILITY FOR NON-CHEMOTHERAPY COHORT: Patients must have met eligibility criteria for the screening step ELIGIBILITY CRITERIA FOR PREREGISTRATION (STEP ): RE-REGISTRATION ELIGIBILITY CRITERIA (STEP ): Patients randomized to Arm may opt to switch to the -drug regimen following disease progression; these patients must be re-registered to the study and meet the eligibility criteria below RE-REGISTRATION ELIGIBILITY CRITERIA (STEP ): Note: G-CSF and platelet transfusions cannot be used to increase counts to meet eligibility criteria ELIGIBILITY CRITERIA (STEP ) ELIGIBILITY CRITERIA - PHASE II (ARM D): Patient must not have Burkitts lymphoma/leukemia based on the WHO criteria ELIGIBILITY FOR SCREENING CRITERIA FOR DONOR ELIGIBILITY: ELIGIBILITY CRITERIA FOR ENROLLMENT Prior eligibility for and on study treatment from an antecedent vemurafenib protocol ELIGIBILITY CRITERIA AT TIME OF TREATMENT: EBV positive tumor ELIGIBILITY CRITERIA AT TIME OF TREATMENT: Patients should have been off other investigational therapy for days prior to infusion MANUFACTURING SJCAR: Meets eligibility criteria to undergo autologous apheresis, or have previously undergone autologous apheresis Eligibility for Infusion of Investigational Product:\r\n* Subjects will undergo a second evaluation of eligibility on day - or - prior to infusion of anti-CD CAR-T cell product. This eligibility criterion will include the inclusion and exclusion criteria required for enrollment with the following exceptions and additions.\r\n** Inclusion criteria exceptions: Hematologic function parameters will not be included as a pre-infusion eligibility criterion (because lymphodepletive chemotherapy is expected to cause pancytopenia). Eligibility for Infusion of Investigational Product:\r\n* Subjects will undergo a second evaluation of eligibility on day - or - prior to infusion of anti-CD CAR-T cell product. This eligibility criterion will include the inclusion and exclusion criteria required for enrollment with the following exceptions and additions.\r\n* Exclusion criteria additions:\r\n** Use of corticosteroids within days prior to infusion (with exception of agents used for prevention of emesis during lymphodepletive chemotherapy).\r\n** Neurologic symptoms suggestive of an active central nervous system condition.\r\n** Signs or laboratory markers of active infection or systemic inflammatory response. Participants must meet appropriate molecular eligibility criteria ADDITIONAL COHORT ELIGIBILITY CRITERIA ADDITIONAL COHORT ELIGIBILITY CRITERIA ELIGIBILITY TO PROCEED WITH PERIPHERAL BLOOD MONONUCLEAR CELL (PBMC) COLLECTION ELIGIBILITY TO PROCEED WITH RICKHAM PLACEMENT\r\n* Once research participants meet eligibility to proceed with Rickham placement, they will be deemed accrued on to the study ELIGIBILITY TO PROCEED WITH CAR T CELL INFUSION ELIGIBILITY CRITERIA FOR ENROLLMENT ON THE RE-TREATMENT STUDY ELIGIBILITY FOR LYMPHODEPLETION CHEMOTHERAPY AND T CELL INFUSION STUDY TREATMENT ELIGIBILITY PRE-ASCT ELIGIBILITY CRITERIA ELIGIBILITY CRITERIA TO BEGIN CONSOLIDATION THERAPY CROSS-OVER ELIGIBILITY CRITERIA Transfusions intended to elevate any parameters below solely for the intent of meeting study eligibility are not permitted REGISTRATION ELIGIBILITY CRITERIA Most recent HR and HER receptor testing should be used to determine eligibility. REGISTRATION ELIGIBILITY ELIGIBILITY CRITERIA FOR nEGFR TESTING ELIGIBILITY CRITERIA FOR STUDY THERAPY PHASE I STUDY ELIGIBILITY CRITERIA:\r\nAll patients must have evaluable disease; biomarker-only disease is not considered evaluable; eligibility of breast cancer with bone only disease is a principal investigator (PI) decision on an individual patient basis PHASE I STUDY ELIGIBILITY CRITERIA:\r\nAbsolute neutrophil count >= ,/mcL PHASE I STUDY ELIGIBILITY CRITERIA:\r\nPlatelets >= ,/mcL PHASE I STUDY ELIGIBILITY CRITERIA:\r\nPatients who were treated with both olaparib and cediranib, either in combination or sequentially ELIGIBILITY CRITERIA- ENROLLMENT AND HARVEST ELIGIBILITY CRITERIA- HARVEST ELIGIBILITY CRITERIA- LYMPHODEPLETION/INFUSION OF tvs-CTL: Autologous transduced peripheral blood T-cells with >= % expression of ga.zeta ELIGIBILITY CRITERIA- LYMPHODEPLETION/INFUSION OF tvs-CTL: ECOG performance score of or less ELIGIBILITY CRITERIA- LYMPHODEPLETION/INFUSION OF tvs-CTL: Absolute neutrophil count > mcL ELIGIBILITY CRITERIA- LYMPHODEPLETION/INFUSION OF tvs-CTL: Platelet > , mcL ELIGIBILITY CRITERIA- LYMPHODEPLETION/INFUSION OF tvs-CTL: Serum AST < x institutional upper limit of normal (IULN) ELIGIBILITY CRITERIA- LYMPHODEPLETION/INFUSION OF tvs-CTL: Total bilirubin < x IULN ELIGIBILITY CRITERIA- LYMPHODEPLETION/INFUSION OF tvs-CTL: Serum creatinine < x IULN ELIGIBILITY CRITERIA- LYMPHODEPLETION/INFUSION OF tvs-CTL: Must not be currently receiving any investigational drugs ELIGIBILITY CRITERIA- LYMPHODEPLETION/INFUSION OF tvs-CTL: Must not have a tumor potentially causing airway obstruction ELIGIBILITY CRITERIA- LYMPHODEPLETION/INFUSION OF tvs-CTL: Must not have received a tumor vaccine within previous six weeks ELIGIBILITY CRITERIA- LYMPHODEPLETION/INFUSION OF tvs-CTL: Must not have a known hypersensitivity to rat monoclonal antibodies ELIGIBILITY CRITERIA- LYMPHODEPLETION/INFUSION OF tvs-CTL: Must not have had a coma or long or multiple seizures within days after a dose of DTP or Tdap unless a cause other than the vaccine was indicated ELIGIBILITY CRITERIA- LYMPHODEPLETION/INFUSION OF tvs-CTL: Must not have evidence of previous or current infection with hepatitis B virus DONOR ELIGIBILITY CRITERIA: Hemoglobin > . g/dL; may transfuse to meet eligibility ELIGIBILITY CRITERIA FOR T-CELL PRODUCT INFUSION: Normal serum sodium levels without need for supplementation ELIGIBILITY CRITERIA FOR T-CELL PRODUCT INFUSION: ALT (SGPT): =< x ULN ELIGIBILITY TO PROCEED WITH PERIPHERAL BLOOD MONONUCLEAR CELL (PBMC) COLLECTION ELIGIBILITY TO PROCEED WITH RICKHAM PLACEMENT ELIGIBILITY FOR ENROLLMENT AND TO PROCEED WITH CAR T CELL INFUSION ELIGIBILITY TO PROCEED WITH PERIPHERAL BLOOD MONONUCLEAR CELL (PBMC) COLLECTION ELIGIBILITY TO UNDERGO LYMPHODEPLETION: (ELIGIBILITY CRITERIA AT TIME OF INFUSION OF GENETICALLY MODIFIED AUTOLOGOUS T CELLS): Research participant has completed prescribed lymphodepletion (ELIGIBILITY CRITERIA AT TIME OF INFUSION OF GENETICALLY MODIFIED AUTOLOGOUS T CELLS): Not requiring pressor support, not having symptomatic cardiac arrhythmias (ELIGIBILITY CRITERIA AT TIME OF INFUSION OF GENETICALLY MODIFIED AUTOLOGOUS T CELLS): Preservation of renal function, serum creatinine did NOT increase by more than fold above the normal range (ELIGIBILITY CRITERIA AT TIME OF INFUSION OF GENETICALLY MODIFIED AUTOLOGOUS T CELLS): Total bilirubin =< . mg/dL ELIGIBILITY TO PROCEED WITH PBMC COLLECTION ELIGIBILITY CRITERIA AT TIME OF INFUSION OF GENETICALLY MODIFIED T CELLS:\r\n* Please note that none of these criteria are applicable of the research participant's donor is undergoing leukapheresis ELIGIBILITY CRITERIA TO UNDERGO OPTIONAL T CELL ABLATION:\r\n* Please note that none of these criteria are applicable of the research participant's donor is undergoing leukapheresis All subjects meeting eligibility criteria irrespective of gender, minority or other underrepresented status will be eligible for enrollment into the study The protocol chairman will determine the eligibility of patients related to hepatic abnormalities If total bilirubin is =< , fractionation is not required for eligibility determination Eligibility criteria cannot be waived ELIGIBILITY CRITERIA FOR REGISTRATION: ECOG performance status =< TREATMENT ELIGIBILITY: TREATMENT ELIGIBILITY: Positive Ga-PSMA-R PET/CT scan for central eligibility assessment. Patients who receive Ga-PSMA-R as part of separate clinical protocol are eligible (must meet all study eligibility criteria) Subjects with a history of autologous stem cell transplant are eligible for study participation provided the following eligibility criteria are met: transplant was > days prior to study enrolment; no active infection(s); subjects meets the remainder of the eligibility criteria outlined in this protocol. ADDITIONAL INDUCTION ELIGIBILITY CRITERIA: GENERAL ELIGIBILITY CRITERIA ELIGIBILITY CRITERIA TO INITIATE TREATMENT (FOR SECONDARY REGISTRATION - COHORT PARTICIPANTS ONLY): ELIGIBILITY CRITERIA: TREATMENT PROTOCOL ELIGIBILITY CRITERIA PRIOR TO FIRST VACCINATION SECONDARY ELIGIBILITY CRITERIA FOR GENE-MODIFIED HSPC INFUSION ELIGIBILITY CRITERIA FOR HSPC TRANSPLANTATION ELIGIBILITY TO RECEIVE T CELL INFUSION: ELIGIBILITY CRITERIA FOR SECOND CELL INFUSION: Subjects must meet all the initial eligibility criteria except for the requirement regarding previous anti-GD-CAR therapy Eligibility for pre-selected salvage chemotherapy, according to the Investigator's assessment. Have had chemotherapy or investigational therapy, with the exception of hydroxyurea, within weeks of study entry; previous treatment at any time with either ruxolitinib or decitabine as single agents will not exclude eligibility; previous stem cell transplant will also not exclude eligibility as long as other inclusion/exclusion criteria have been met Once all other eligibility criteria are confirmed, must have an apheresis product of non-mobilized cells received and accepted by the manufacturing site. Note: Apheresis product will not be shipped to or assessed for acceptance by the manufacturing site until documented confirmation of all other eligibility criteria is received. PROSPECTIVE SCREENING ELIGIBILITY CRITERIA: REGISTRATION (POST-SURGERY) ELIGIBILITY CRITERIA ALT < ULN NOTE: Laboratory results obtained during screening should be used to determine eligibility criteria. In situations where laboratory results are outside the permitted range, the investigator may retest the subject and the subsequent within range screening result may be used to determine the subject's eligibility. REGISTRATION ELIGIBILITY CRITERIA Patient meets the eligibility criteria outlined above CONSOLIDATION ELIGIBILITY: Any individual that does not meet the eligibility criteria for transplantation or donor eligibility will not be a part of this trial Persistence of clinically relevant therapy related toxicity from previous chemotherapy and/or radiotherapy; this does not include hemoglobin or other hematologic or laboratory criteria, as long as eligibility criteria are met as outlined above Patients may have had enucleation of one eye, as long as the remaining eye meets the eligibility criteria ELIGIBILITY CRITERIA AT TIME OF INFUSION OF GENETICALLY MODIFIED AUTOLOGOUS T CELLS: Exceptions to eligibility will not be granted for this study REGISTRATION ELIGIBILITY CRITERIA PHASE I PORTION ELIGIBILITY CRITERIA ELIGIBILITY TO PROCEED TO OVA: FURTHER ELIGIBILITY DETAILS FOR PATIENTS WITH OPERABLE DISEASE (COHORT ): REGISTRATION ELIGIBILITY ELIGIBILITY PRIOR TO CELL COLLECTIONS FOR DENDRITIC CELL GENERATION: RANDOMIZATION ELIGIBILITY CRITERIA In the event of significant BM involvement, the above hematologic criteria will not be required for enrollment eligibility The patient currently does not meet the protocols eligibility/enrollment criteria for any reason Patient must meet eligibility criteria for allogeneic transplantation ELIGIBILITY CRITERIA FOR HISTORICAL CONTROL POPULATION PRIMARY ELIGIBILITY (PRE-OPERATIVE [OP]) General Eligibility Criteria (All Parts) Subjects must meet general eligibility criteria. The specific eligibility criteria listed here will apply to subjects enrolling to different cohorts of Part B. Specific Eligibility Criteria, Part C - Subjects must meet general eligibility criteria. Specific Eligibility Criteria, Part D - Subjects must meet general eligibility criteria Patients may not have clinically symptomatic hypothyroidism; testing is not required for eligibility ELIGIBILITY PRIOR TO INITIATING CT-: STEP ELIGIBILITY CRITERIA STEP ELIGIBILITY CRITERIA Patients who had prior lung resection are eligible provided they fulfill the rest of the eligibility criteria REGISTRATION/RANDOMIZATION ELIGIBILITY CRITERIA Centers that standardly use positron emission tomography (PET) or magnetic resonance spectroscopy (MRS) to determine a diagnosis of radionecrosis are permitted to use these modalities to assist in their patient selection; however the criteria described for conventional MR and/or DSC should also be met for study eligibility; both PET and MRS are not mandatory for study eligibility Central imaging real-time review ( hour turn around) to confirm eligibility (for institutions that opt to utilize central imaging review to confirm eligibility) Patient eligibility criteria for entry into the project include: FCG eligibility criteria include: PATIENT ELIGIBILITY CRITERIA FCG has been invited to participate in the trial with a patient who meets eligibility criteria ELIGIBILITY CRITERIA FOR BOTH SURVIVORS AND CO-SURVIVORS: The patient is enrolled on a COG trial that uses criteria for unrelated donor HSCT, which conflict with our eligibility criteria PATIENTS ELIGIBILITY CRITERIA FOCUS GROUP (PHASE ) ELIGIBILITY CRITERIA: USER/USABILITY TESTING (PHASE ) ELIGIBILITY CRITERIA: PATIENTS ELIGIBILITY CRITERIA: PATIENT PARTICIPANT ELIGIBILITY CRITERIA (PHASE & ) CLINICIAN PARTICIPANT ELIGIBILITY CRITERIA STAKEHOLDER PARTICIPANT ELIGIBILITY CRITERIA PATIENT ELIGIBILITY REQUIREMENTS: Eligibility will not be restricted by race or sex Participants must have already met one or more eligibility criteria and have a reasonable expectation of meeting any remaining eligibility criteria for the therapeutic clinical trial PATIENT REGISTRATION ELIGIBILITY CRITERIA: ELIGIBILITY FOR THE OPTIONAL SUB-STUDY ELIGIBILITY FOR THE -YEAR EXTENSION GENERAL ELIGIBILITY (ALL PATIENTS): ELIGIBILITY FOR REDUCED INTENSITY CONDITIONING: Completed all eligibility questions Subjects with a history of autologous stem cell transplant are eligible for study participation provided the following eligibility criteria are met: transplant was > days prior to study enrolment, no active infection; subject meets the remainder of the eligibility criteria outlined in the study protocol. ELIGIBILITY CRITERIA FOR NORMAL-WEIGHT WOMEN IN PILOT STUDY A pregnancy test will be used to determine eligibility in appropriate patients STEP : ELIGIBILITY CRITERIA PRIOR TO TREATMENT OR OBSERVATION Required patient clinical data is not available for evaluation of eligibility criteria PRIMARY ELIGIBILITY (PRE-OPERATIVE [OP]) PART A ELIGIBILITY CRITERIA PART B ELIGIBILITY CRITERIA REGISTRATION ELIGIBILITY CRITERIA