No intent to use myeloablative conditioning regimens. Patient deemed to be suitable candidate for myeloablative or reduced intensity conditioning allogeneic HSCT using PBSC or marrow as stem cell source Participants must be recipients of an allogeneic bone marrow or stem cell transplantation with myeloablative or reduced intensity conditioning regimens Have undergone allogeneic stem cell transplantation (alloSCT) from any donor source (matched unrelated donor, sibling, haplo-identical) using bone marrow, peripheral blood stem cells, or cord blood. Recipients of non-myeloablative, myeloablative, and reduced intensity conditioning are eligible Participant is considered a suitable candidate for HCT and has an acceptable source of allogeneic donor stem cells, as defined per institutional practice (allogeneic HCT for any donor source [matched sibling, unrelated donor (URD), mismatched URD, related haploidentical, or umbilical cord blood] and any graft source [umbilical cord, BM, peripheral blood (PB)], and any conditioning [myeloablative conditioning (MAC), reduced intensity conditioning (RIC), or non-myeloablative conditioning (NMA)] will be permitted). 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 investigators 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 Gray [Gy]) Any prior myeloablative transplant within the last months Patients who have received Y- ibritumomab (Zevalin) or I- tositumomab (Bexxar), as part of their salvage therapy are not eligible for myeloablative umbilical cord blood transplant 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 >= years and < 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 Have undergone Allogeneic Stem Cell Transplanttaion (alloSCT) from any donor source (matched unrelated donor, sibling, haplo-identical) using bone marrow, peripheral blood stem cells, or cord blood. Recipients of non- myeloablative, myeloablative, and reduced intensity conditioning are eligible Deemed to be not otherwise eligible for a myeloablative hematopoietic cell transplant; high risk characteristics for a myeloablative transplant include age > years and a HCT comorbidity index > Less than months since prior myeloablative transplant Patients with HM who have undergone myeloablative systemic therapy are ineligible to participate in this study. Must be >= months after prior myeloablative transplant, if applicable Patients with or without previous myeloablative autologous transplant Any donor type (e.g., related, unrelated) or stem cell source (bone marrow, peripheral blood, cord blood); recipients of non-myeloablative and myeloablative transplants are eligible 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 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 STEP : Within days of study entry: Patients with a history of bone marrow disorders including hematological malignancies and aplastic anemia, myelodysplastic Syndrome (MDS) and myeloproliferative disorder (MPD) planning to undergo allogeneic HSCT with reduced intensity or myeloablative conditioning regimens 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 months Must receive a myeloablative or reduced intensity conditioning regimen for stem cell transplant (SCT) as defined by the Center for International Blood and Bone Marrow Transplant Research (CIBMTR)\r\n* Cyclophosphamide (Cy) and single dose total body irradiation\r\n* Fludarabine (Flu) and busulfan\r\n* Fractionated total body irradiation (TBI) and cyclophosphamide\r\n* Busulfan and cyclophosphamide Patients developing aGvHD after ablative or non-myeloablative or reduced intensity conditioning will be eligible Myeloablative transplant within the last months If > years old, prior myeloablative transplant within the last months If =< years old, prior myeloablative transplant within the last months; if > years old prior myeloablative allotransplant or autologous transplant Patients who have received Y- ibritumomab (zevalin) or I- tostumomab (bexxar), as part of their salvage therapy are not eligible for myeloablative umbilical cord blood transplant Second BMT: must be >= months after prior myeloablative transplant If =< years old, prior myeloablative transplant within the last months; if > years old prior myeloablative allotransplant or autologous transplant Patients are eligible 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)- ibritumomab (Zevalin) or iodine (I)- 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 >= 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 Recipients of -/ 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 Gy) Myeloablative or non-myeloablative allogeneic hematopoietic cell transplant Prior myeloablative transplant within previous months of study enrollment Prior allogeneic or autologous stem cell transplant using a myeloablative busulfan or total body radiation containing conditioning regimen defined as busulfan-based using a total dose of >= mg/kg given by mouth or >= mg/kg given IV; or a total-body irradiation (> Gy) Standard pediatric indications for myeloablative transplantation for patients undergoing HSCT at Childrens National Medical Center per institutional guidelines Age greater than or equal to year but less than or equal to years (myeloablative regimen ); eligibility for pediatric patients will be determined in conjunction with an MD Anderson Cancer Center (MDACC) pediatrician; patients > but < years who have a performance status of or and no comorbidities may receive the myeloablative regimen at the discretion of the investigator(s) Prior myeloablative transplant within the last 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 months. Patients are eligible 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 > year post transplant, have been off all immunosuppressive therapy for more than months and do not have active graft-versus-host disease (GVHD) Patients are eligible 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 I-MIBG) are eligible at any time as long as they meet the other criteria for eligibility. Donor cellular engraftment of at least .% from the non-myeloablative procedure and prior to the first infusion Subjects who had histopathologically confirmed overall grade GVHD lasting longer than 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 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 =< % 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 >= months post initial transplant, achieve a CR or have =< % 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) >= cGy, or \r\n* Busulfan >= . 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 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) Patients designated to undergo myeloablative or intermediate intensity allogeneic peripheral blood or bone marrow hematopoietic cell transplantation; consent will be obtained prior to admission for HSCT; patients receiving any donor source of stem cells are eligible; eligible conditioning regimens are those defined as myeloablative by the American Society of Blood and Marrow Transplantation (ASBMT) consensus criteria as well as the combination of fludarabine with melphalan (- mg/mg^) Recipients of allogeneic hematopoietic cell transplantation (HCT) after either myeloablative or reduced intensity conditioning regimens; any donor source of stem cells is eligible All conditioning regimens will be eligible (in case of allogeneic HCT, patient could have received myeloablative or non-myeloablative/reduced-intensity conditioning) Recipients of allogeneic hematopoietic cell transplantation (HCT) after either myeloablative or reduced intensity conditioning regimens; any donor source of stem cells is eligible Prior myeloablative or non-myeloablative allogeneic hematopoietic stem cell transplant using either bone marrow or peripheral blood stem cells within months Myeloablative preparative regimen (i.e., >= TBI . Gy, >= busulfan (BU) . mg/kg PO, >= BU . mg/kg intravenously (IV), >= treosulfan g/m^ 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 months from myeloablative conditioning for autologous transplantation Planned myeloablative conditioning regimen Patients who have undergone a non-myeloablative stem cell transplant must have > % donor hematopoiesis within days of study enrollment; chimerism within days of study entry must be greater than, equal to, or no more than % less than the chimerism measured at approximately day + (if performed) Patients who have undergone a non-myeloablative stem cell transplant must have > % donor lymphoid hematopoiesis within days of study enrollment ELIGIBILITY FOR MYELOABLATIVE CONDITIONING Myeloablative preparative regimen Less than months from myeloablative conditioning for autologous transplantation (if applicable) Be scheduled to receive one of myeloablative conditioning regimens (defined in population) followed by allogeneic SCT for hematological malignancy.