Prior autologous or allogeneic HCT Patients should be deemed to be potentially eligible and willing candidates for auto-HCT by the enrolling physician Ages > years with hematologic malignancies treatable by related or unrelated HCT Ages =< years of age with hematologic diseases treatable by allogeneic HCT who refuse a high-dose HCT; transplants must be approved for these inclusion criteria by the principal investigators at the collaborating centers and at FHCRC Previous history of autologous or allogeneic-HCT A prospective patient for allogeneic HCT for a malignant hematologic disorder Prior cytotoxic chemotherapy within days from the initiation of HCT conditioning. Note, certain low intensity treatments not intended to induce remission but rather stabilize disease are acceptable up to days prior to initiation of HCT conditioning Prior allogeneic or autologous HCT at any time. Prior allogeneic HCT. Patients - years old will be enrolled in Arm B or D (lower intensity); patients eligible for Arms B or D also include those who have received previous allogeneic HCT, or who have co-morbid conditions rendering them unsuitable for high-dose conditioning, determined in consultation with the principal investigator PHASE I: If post allogeneic HCT: Confirmed CD+ leukemia recurrence defined as >= .% disease by Seattle Childrens Hospital (SCH) or University of Washington (UW) Pathology Department following allogeneic HCT Patients with a nonmalignant disease treatable by allogeneic HCT AUTOLOGOUS APHERESIS: Patients with a history of prior allogeneic hematopoietic cell transplantation (HCT) must be clinically recovered from prior HCT therapy, have no evidence of active graft versus host disease (GVHD) and have not received a donor lymphocyte infusion (DLI) within the days prior to apheresis MANUFACTURING SJCAR: CD+ ALL with any of the following:\r\n* Primary refractory disease despite at least cycles of an intensive chemotherapy regimen designed to induce remission\r\n* Refractory disease despite salvage therapy\r\n* nd or greater relapse\r\n* Any relapse after allogeneic hematopoietic cell transplantation \r\n* st relapse if patient requires an allogeneic HCT as part of standard of care relapse therapy, but is found to be ineligible and/or unsuitable for HCT The use of serotherapy to prevent GVHD (e.g., antithymocyte globulin) prior to day post-HCT is permitted. Patients with a prior autologous or allogeneic HCT Have received a T cell-depleted allogeneic (i.e., non-autologous) HCT within the previous days. If post allogeneic HCT: Confirmed CD+CD+ leukemia recurrence defined as >= .% disease following allogeneic HCT If relapse/refractory status with no prior history of allogeneic HCT (one of the following)\r\n* nd or greater marrow relapse, with or without extramedullary disease\r\n* st marrow relapse at end of st month of re-induction with marrow having >= .% blast by morphology and/or MPF\r\n* Primary refractory as defined as having > % blasts by multi-parameter flow or polymerase chain reaction (PCR) after or more separate induction regimens\r\n* Subject has indication for HCT but has been deemed ineligible, inclusive of persistent MRD prior to HCT Patients who have relapsed or have MRD after HCT may receive other agents for treatment of disease and remain eligible for the protocol Previous HCT Has had relapse prior to primary neutrophil engraftment or =< days post HCT Ages > years with hematologic malignancies treatable by unrelated hematopoietic cell transplant (HCT) Ages =< years with hematologic diseases treatable by allogeneic HCT who refuse a high-dose HCT; transplants must be approved for these inclusion criteria by the principal investigators at the collaborating centers and at FHCRC If post allogeneic HCT: confirmed CD+ leukemia recurrence defined as >= .% disease by following allogeneic HCT Prior allo-HCT less than three months from the time of enrollment Received an allogeneic HCT within the last days; enrollment within - days after transplant, and after adequate recovery of counts Prior autologous or allogeneic HCT Prior allogeneic HCT RESEARCH SAMPLE COLLECTION: Considered for high-dose melphalan followed by autologous hematopoietic cell transplantation and undergoing pre-HCT workup RESEARCH SAMPLE COLLECTION: Myeloma patients who are not candidates for high-dose melphalan followed by autologous HCT based on institutional standards Received high-dose melphalan (>= mg/m^) followed by autologous HCT based on the institutional guidelines and within + and + after autologous HCT at the time of panobinostat maintenance initiation Prior allogeneic HCT Prior allogeneic HCT Subjects with second or subsequent relapse, any relapse refractory to salvage, or with persistent disease after at least two lines of therapy\r\n* Subjects with relapsed disease after prior allogeneic HCT (myeloablative or non-myeloablative) will be eligible if they meet all other inclusion criteria and\r\n** Have experienced graft rejection (no evidence of donor cells by short tandem repeat [STR] analysis on occasions separated by at least month), OR\r\n** Donor cells are present but there is no active graft-versus-host disease (GVHD), subject does not require immunosuppression and is more than months from transplant\r\n** Have relapsed after prior syngeneic HCT and is more than months from transplant\r\n* Subjects with relapsed disease after prior autologous HCT will be eligible if they meet all other inclusion criteria and it has been more than months from transplant Expected to undergo HCT within days of enrollment Does not have any other active malignancy other than the one for which this HCT is indicated No prior allogeneic HCT, and no autologous HCT within the previous months Patients who are currently undergoing or who previously underwent matched allogeneic HCT for:\r\n* AML: Prospective enrollment will now be limited to patients with relapsed disease (overt relapse or minimal residual disease) at any time post allogeneic HCT\r\n* MDS will no longer be a criterion for eligibility\r\n* CML will no longer be a criterion for eligibility Allowed prior allogeneic hematopoietic stem cell transplantation (allo-HCT) regardless of stem cell source; patients must be at least months post allo-HCT (at time of treatment start); mismatched transplantations would be allowed If post allo-HCT, then patient must have baseline donor T cell chimerism of >= % (from peripheral blood); evaluation can be made within weeks of treatment start Active bacterial infection within one week of HCT Active fungal infection at time of HCT First autologous or allogeneic HCT and hematologic disease in remission on initiation of antiviral therapy for hepatitis C infection If undergoing myeloablative allogeneic HCT: If undergoing non-myeloablative allogeneic HCT: Ongoing systemic immunosuppressive therapy after haploidentical HCT Administration of G-CSF after haploidentical HCT CD+ cells ? /l at day of planned experimental infusion after haploidentical HCT Patients with prior allogeneic hematopoietic cell transplantation (HCT) must be at least days post-HCT and must be on =< mg of prednisone (or equivalent dose of an alternative corticosteroid) for treatment/prevention of graft-vs-host disease Patients with previous autologous or allogeneic HCT are allowed to enroll > % blast in the PB or BM prior to hematopoietic cell transplantation (HCT) or had leukemic transformation (> % blasts in PB or BM any time prior to HCT) Patient willing to consider HCT Autologous HCT < months prior to the time of signing the informed consent form Prior allogeneic HCT Have previously received HCT Planning to receive autologous HCT per institutional standards as part of standard of care. Eligibility for autologous HCT should be based on institutional guidelines. However, at minimum all patients must meet the following criteria:\r\n* Karnofsky performance status (KPS) greater than \r\n* Cardiac left ventricular ejection fraction of greater than %\r\n* Calculated creatinine clearance of greater than cc/min\r\n* Aspartate aminotransferase (AST) and alanine aminotransferase (ALT) of less than x upper limit of normal\r\n* Direct bilirubin of less than x upper limit of normal Prior allogeneic HCT (prior autologous transplant is allowed regardless of response) Relapse of primary hematologic malignancy that served as indication for HCT Prior allogeneic HCT. Ages > years with hematologic malignancies treatable by unrelated hematopoietic cell transplant (HCT) Ages =< years of age with hematologic diseases treatable by allogeneic HCT who refuse a high-dose HCT; transplants must be approved for these inclusion criteria by the principal investigators at the collaborating centers and at FHCRC Recurrence or progression of primary malignancy after HCT Patients must be expected to have disease controlled for at least days after HCT Patients who have received a prior allogeneic HCT must have no active GVHD requiring immunosuppressive therapy for at least days prior to start of conditioning Patients will then be assigned to one of two cohorts:\r\n* Cohort will include patients who have relapsed /progressed within the first days post-transplant and who are still within months from date of progression-relapse\r\n* Cohort will include patients who have either i) relapsed/progressed beyond day post-HCT, ii) those with persistent stable disease or persistent disease with regression between days - after allogeneic HCT, or iii) those who progressed or relapsed within days after HCT but were not started on this protocol within months from date of progression or relapse could also be enrolled under cohort \r\n** NOTE: the inclusion of patients with persistent stable or persistent regressing disease in this protocol is not meant to advocate treatment; however, if the attending physician is inclined to offer treatment then these patients would be eligible for this study No evidence of HCT graft failure or multi-organ failure Candidate committed to HCT independent of participation in this study, with the following requirements: Meets local transplant center eligibility requirements for HCT Prior allogeneic or autologous HCT or adoptive cellular therapies, including T-cell chimeric antigen receptor (CAR) therapy Diagnosis of a nonmalignant disorder considered treatable by HCT. Subjects with prior ASCT or allogenic HCT. Subjects ineligible for available curative options after failing ASCT and have met the hematologic criteria are eligible to participate in this study. Subjects with prior allogenic HCT will be excluded. Extramedullary disease noted during pre-HCT work up can receive a boost of radiation as clinically indicated Patient has any other active malignancy other than the one for which HCT is indicated Anti-thymocyte globulin, alemtuzumab, bortezomib, or cyclophosphamide administered within days before or planned to receive with HCT conditioning or as part of GVHD prophylaxis in the days after HCT Patients must be previously untreated with HCT Normal WBC (.-. x L), PLT (- x L), and HCT (-%) Planned use of post-HCT cyclophosphamide for GVHD prophylaxis T deplete HCT Adult caregivers of patients undergoing autologous or allogeneic HCT at Massachusetts General Hospital (MGH), and they must attend the HCT consent visit with the patient Received >= autologous or allogeneic (related or unrelated) HCT with curative intent at a participating transplant center for a hematologic malignancy Survival - years after last HCT when first approached for enrollment Refusing to use contraception up to days post-HCT At least year after HCT Planned HCT with minimal to no-T cell depletion of graft Conditioning regimens days (d) prior to trial participation and up to d post-HCT Patients considered by PIs/protocol team to have a complicated prior therapy or HCT regimen, or who have a low survival probability (e.g., refractory leukemia and/or undergoing nd HCT) Adult patients with hematologic malignancy who underwent an allogeneic HCT at least months prior to study enrollment Patients with relapsed disease post-HCT Planned HCT with no ex-vivo T cell depletion of graft; conditioning and immunosuppressive regimens according to institutional guidelines are permitted Planned HCT with minimal to no-T cell depletion of graft Participants must be at least days after HCT Adult (? years at time of allogeneic HCT recipient at participating transplant centers) Patient may have received more than one HCT Participants must be at least days after HCT Currently - years after first HCT Prior autologous or allogeneic HCT; RECIPIENT: Planned medications from the time of HCT to day post?HCT Between day + and day + after allogeneic HCT Patients must have either relapsed after previous high-dose chemotherapy and autologous or allogeneic HCT, or else be ineligible for such an approach due to age, failure to mobilize sufficient hematopoietic stem cells, medical comorbidities, or patient refusal Patients who refuse to be treated on a conventional autologous or allogeneic HCT protocol Planned haematopoietic stem cell transplant (HCT) during the study period. (If a HCT occurred prior to enrolment in the study, the subject may not receive study vaccine until at least days after the transplant procedure). Patients undergoing MRD allogeneic HCT Patients who have not received an allogeneic HCT At least months after HCT, either autologous or allogeneic (of any source, with any preparatory regimen, for any indication), prior to study treatment. Evidence of active malignancy at the time of HCT or at any time since the HCT. Scheduled to receive an autologous HCT Did not receive an allogeneic HCT at Dana-Farber