No intent to use myeloablative conditioning regimens. Participants must be recipients of an allogeneic bone marrow or stem cell transplantation with myeloablative or reduced intensity conditioning regimens Participants may receive either a myeloablative or a non-myeloablative (reduced-intensity) conditioning regimen Recipient of a myeloablative or non-myeloablative allogeneic hematopoietic stem cell transplant (HSCT):\r\n* Conditioning regimen to be prescribed at investigator’s discretion, but will be prospectively defined as myeloablative or non-myeloablative Patients that have undergone immunotherapy in combination with non-myeloablative chemotherapy Prior myeloablative transplant containing full dose TBI (greater than 8 Gray [Gy]) Any prior myeloablative transplant within the last 6 months Will undergo first allogeneic hematopoietic stem cell transplantation (HSCT) for their malignancy. Conditioning may have been either conventional myeloablative (MAC) or reduced intensity conditioning (RIC) Patients >= 12 years and < 55 years are also included if they are not candidates for myeloablative conditioning regimens due to comorbidities Any conditioning regimen (non-myeloablative, myeloablative, or reduced intensity) is acceptable. Diagnosis: Patients with disorders affecting the hematopoietic system that are inherited, acquired, or result from myeloablative treatment. Use of myeloablative conditioning regimen Recipients of allogeneic stem cell transplantation with myeloablative or non-myeloablative conditioning regimens; alternative donor transplants (umbilical cord blood and haploidentical) are allowed Less than 3 months since prior myeloablative transplant (if applicable); less than 6 months since prior autologous transplant (if applicable) Deemed to be not otherwise eligible for a myeloablative hematopoietic cell transplant; high risk characteristics for a myeloablative transplant include age > 60 years and a HCT comorbidity index > 3 Less than 3 months since prior myeloablative transplant Patients with HM who have undergone myeloablative systemic therapy are ineligible to participate in this study. Must be >= 3 months after prior myeloablative transplant, if applicable Patients with or without previous myeloablative autologous transplant First allogeneic hematopoietic stem cell transplantation (HSCT) using myeloablative conditioning (MAC), non-myeloablative (NMA), or reduced-intensity conditioning (RIC) preparative regimens. Prior myeloablative or non-myeloablative allogeneic hematopoietic stem cell transplant using either bone marrow or peripheral blood stem cells or single or double cord blood within 24 months. Prior myeloablative or non-myeloablative autologous or allogeneic hematopoietic stem cell transplant using the marrow, peripheral blood stem cells or single or double umbilical cord blood Patients ineligible to receive full myeloablative conditioning regimen for allogeneic hematopoietic progenitor cell transplant due to age or comorbidities Prior myeloablative or non-myeloablative allogeneic hematopoietic stem cell transplant using either bone marrow or peripheral blood stem cells or single or double cord blood within 18 months Patients with relapsed disease after prior allogeneic SCT (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 2 occasions separated by at least 1 month)\r\n* Have no active graft-vs-host disease (GVHD) and require no immunosuppression\r\n* Are more than 6 months from transplant Patients developing aGvHD after ablative or non-myeloablative or reduced intensity conditioning will be eligible Second hematopoietic cell transplant: Must be >= 3 months after prior myeloablative transplant Myeloablative transplant within the last 6 months If > 18 years old, prior myeloablative transplant within the last 6 months If =< 18 years old, prior myeloablative transplant within the last 6 months; if > 18 years old prior myeloablative allotransplant or autologous transplant Second BMT: must be >= 3 months after prior myeloablative transplant If =< 18 years old, prior myeloablative transplant within the last 6 months; if >18 years old prior myeloablative allotransplant or autologous transplant Patients are eligible 12 weeks after myeloablative therapy with autologous stem cell transplant (timed from start of vorinostat); patients must meet adequate bone marrow function definition post-myeloablative therapy; patients who received stem cell reinfusion following non-myeloablative therapy are eligible once they meet peripheral blood count criteria Prior myeloablative allotransplant Patients who have received yttrium (Y)-90 ibritumomab (Zevalin) or iodine (I)-131 tositumomab (Bexxar), as part of their salvage therapy are not eligible for myeloablative UCB transplant Myeloablative or non-myeloablative allogeneic hematopoietic cell transplantation No prior myeloablative therapy or hematopoietic cell transplantation Must be >= 3 months after prior myeloablative transplant, if applicable Has received prior allogeneic transplants or who are planned to undergo umbilical cord blood transplant, receive ex vivo T-cell-depleted hematopoietic stem cells (HSCs), received any in vivo T-cell depleting antibodies, or non-myeloablative conditioning. Applicable disease and eligible for myeloablative SCT Patients with relapsed disease after prior allogeneic SCT (myeloablative or non-myeloablative) will be eligible if they meet all other inclusion criteria and\r\n* Have no active graft-versus-host disease (GVHD) and require no immunosuppression\r\n* Are more than 6 months from transplant Recipients of 7-8/8 human leukocyte antigen (HLA) matched adult donor allogeneic stem cell transplantation with myeloablative or non-myeloablative conditioning regimens Patients with uncontrolled infections for whom myeloablative hematopoietic stem cell transplant (HCT) is considered contraindicated by the consulting infectious disease physician (upper respiratory tract viral infection does not constitute an uncontrolled infection in this context) Prior myeloablative transplant containing full dose TBI (greater than 8 Gy) Myeloablative or non-myeloablative allogeneic hematopoietic cell transplant Prior myeloablative transplant within previous 3 months of study enrollment Prior myeloablative transplant within the last 6 months Prior myeloablative or non-myeloablative allogeneic hematopoietic stem cell transplant using either bone marrow or peripheral blood stem cells or single or double cord blood within 24 months. Patients are eligible 12 weeks after myeloablative therapy with autologous stem cell transplant (timed from start of protocol therapy); patients must meet adequate bone marrow function definition post-myeloablative therapy; patients who received stem cell reinfusion following non-myeloablative therapy are eligible once they meet peripheral blood count criteria; patients status post-allogeneic stem cell transplant are excluded unless they are > 1 year post transplant, have been off all immunosuppressive therapy for more than 3 months and do not have active graft-versus-host disease (GVHD) Patients are eligible 12 weeks after date of autologous hematopoietic stem cell infusion following myeloablative therapy (timed from first day of this protocol therapy). Patients who have received an autologous hematopoietic stem cell infusion to support non-myeloablative therapy (such as 131I-MIBG) are eligible at any time as long as they meet the other criteria for eligibility. Donor cellular engraftment of at least 2.5% from the non-myeloablative procedure and prior to the first infusion Subjects who had histopathologically confirmed overall grade 4 GVHD lasting longer than 7 days, from the non-myeloablative therapy, are not eligible Acute lymphoblastic leukemia (ALL) \r\n * ALL patients will be eligible if they fail to attain an initial remission, if they relapse within 1 year following the discontinuation of chemotherapy, or if they have other unfavorable prognostic features such that a stem cell transplant (SCT) would offer a significant survival advantage; patients must be in complete remission or have =< 25% blasts in bone marrow at the time of admission to the HSCT unit; patients in complete remission will preferentially receive a myeloablative transplant from a related or unrelated donor; however, patients will be eligible for this study if a suitable related or unrelated donor cannot be identified, the amount of time required to identify a suitable donor is deemed unacceptable, or the patient is not eligible for a myeloablative transplant regimen\r\n * Patients who relapse following a myeloablative transplant, but cannot receive DLI (e.g. cord blood recipients, graft loss) will also be eligible; such patients must be >= 6 months post initial transplant, achieve a CR or have =< 25% blasts in the bone marrow prior to admission to the HSCT unit Meet standard criteria as defined by the institution for a myeloablative allogeneic stem cell transplantation, with myeloablative defined as using conditioning regimens containing:\r\n* Total-body irradiation (TBI) >= 1200 cGy, or \r\n* Busulfan >= 12.8 mg/kg Myeloablative preparative regimen (for SAA any conditioning therapy allowed) Conditioning Regimen: Patients expecting to receive any type of myeloablative HSCT conditioning regimen are eligible Between day +28 and day +42 status post myeloablative or nonmyeloablative UCBT (single or double cord blood transplant) Patients must be receiving a fractionated total body radiation (FTBI) based- myeloablative conditioning regimen; (acceptable conditioning regimens include total body irradiation [TBI] + cyclophosphamide or TBI + etoposide) All conditioning regimens will be eligible (in case of allogeneic HCT, patient could have received myeloablative or non-myeloablative/reduced-intensity conditioning) Prior myeloablative or non-myeloablative allogeneic hematopoietic stem cell transplant using either bone marrow or peripheral blood stem cells within 12 months Myeloablative preparative regimen (i.e., >= TBI 12.0 Gy, >= busulfan (BU) 8.0 mg/kg PO, >= BU 6.4 mg/kg intravenously (IV), >= treosulfan 42 g/m^2 IV) according to investigational study or standard treatment plan; other \myeloablative\ preparative regimens are acceptable as long as they are approved by the principal investigator or designee Less than 3 months from myeloablative conditioning for autologous transplantation Planned myeloablative conditioning regimen Patients who have undergone a non-myeloablative stem cell transplant must have > 80% donor hematopoiesis within 30 days of study enrollment; chimerism within 30 days of study entry must be greater than, equal to, or no more than 5% less than the chimerism measured at approximately day +30 (if performed) Patients who have undergone a non-myeloablative stem cell transplant must have > 65% donor lymphoid hematopoiesis within 30 days of study enrollment ELIGIBILITY FOR MYELOABLATIVE CONDITIONING Myeloablative preparative regimen Less than 3 months from myeloablative conditioning for autologous transplantation (if applicable) Be scheduled to receive one of 3 myeloablative conditioning regimens (defined in population) followed by allogeneic SCT for hematological malignancy.