Has undergone 1 allo-HSCT from any donor (related or unrelated with any degree of HLA matching) and any donor source (bone marrow, peripheral blood stem cells, or cord blood) for a hematologic malignancy or disorder. Recipients of myeloablative and reduced-intensity conditioning regimens are eligible. Patient deemed to be suitable candidate for myeloablative or reduced intensity conditioning allogeneic HSCT using PBSC or marrow as stem cell source DONOR: Only filgrastim (G-CSF) mobilized PBSC only will be permitted as a hematopoietic stem cell (HSC) source on this protocol DONOR: Bone marrow will be the only allowed stem cell source 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 Patients must have a related or unrelated peripheral blood stem cell donor that meet one of the following criteria: 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). DONOR: must be capable of and consent or assent to donation of peripheral blood stem cells Patient must have an adequate supply of stem cells for transplant harvested prior to study enrollment, with adequate supply defined as 3 x 10^6 cluster of differentiation (CD)34+ cells/kg for peripheral blood stem cells (PBSC); cell mobilization method will be left up to the treating physician's discretion and may include mobilization growth factor alone or mobilization after chemotherapy; if patient is unable to mobilize the proper amount of peripheral stem cells, bone marrow may be harvested as the source of hematopoietic stem cells; in this instance, 3 x 10^8 mononuclear cells/kg will be considered adequate; if necessary, a combination of peripheral stem cells and bone marrow can be used Any donor type (e.g., related, unrelated) or stem cell source (bone marrow, peripheral blood, cord blood). PHASE I: Patients who underwent an allogeneic hematopoietic stem cell transplantation from any donor source for acute lymphocytic leukemia PHASE II: Patients who underwent an allogeneic hematopoietic stem cell transplantation from any donor source for acute lymphocytic leukemia Have identified a backup cells source in case of engraftment failure. The source can be autologous, related or unrelated. 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 A person (such as a haploidentical family member) or unit of cord blood must be identified as a source of back-up cells source in case of engraftment failure 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 Hematopoietic stem cell source is either with peripheral blood, bone marrow or cord blood. No more than 1 antigen mismatch at HLA-A, -B, -C, -DRB1 or -DQB1 locus for unrelated donor with peripheral blood and bone marrow as the hematopoietic stem cell source; and DONOR: Donors >= 18 years of age must be the same individual whose cells were used as the source for the patient’s original stem cell transplant Marrow is the preferred source of stem cells from the HLA-haploidentical donor, however, peripheral blood mononuclear cells (PBMC) could be used as stem cell source, after clearance with the Fred Hutchinson Cancer Research Center (FHCRC) principal investigator, in the case of difficulties or contraindications to bone marrow harvest from the donor DONOR: Marrow is the preferred source of stem cells from the HLA-haploidentical donor, however PBMC could be used as stem cell source, after clearance with the FHCRC principal investigator, in the case of difficulties or contraindications to bone marrow harvest from the donor Stem cell source Bone marrow (preferred source) OR peripheral blood stem cells of filgrastim [G-CSF]-stimulated donors OR cord blood Have identified a backup cells source in case of engraftment failure; the source can be autologous, related or unrelated Donor stem cell source can be either peripheral blood or bone marrow Cord blood as a donor source is not acceptable Have undergone first allogeneic hematopoietic stem cell transplantation (allo-HSCT) from any donor source using bone marrow, peripheral blood stem cells, or cord blood for hematologic malignancies. Recipients of nonmyeloablative and myeloablative conditioning regimens are eligible. HLA Matched Related Donor: G-CSF mobilized peripheral blood mononuclear cell (PBMC) only will be permitted as a hematopoietic stem cell (HSC) source on this protocol Back-up stem cell source DONOR: Peripheral blood stem cells (PBSC) only will be permitted as a HSC source on this protocol Have undergone first allo-HSCT from any donor source (matched unrelated donor, sibling, haploidentical) using bone marrow, peripheral blood stem cells, or cord blood for hematologic malignancies. Recipients of nonmyeloablative and myeloablative transplants are eligible. DONOR: Meets standard institutional criteria for GCSF mobilized peripheral blood stem cell (PBSC) donation Use of cord blood as the source of hematopoietic cells is not allowed. DONOR: Only filgrastim (G-CSF) mobilized PBSC only will be permitted as a hematopoietic stem cell (HSC) source on this protocol DONOR: Marrow will be the only allowed hematopoietic stem cell source Unrelated cord blood will be used as a source of hematopoietic support if a 5 or 6/6 related or 6/6 unrelated bone marrow donor is not available, or if the tempo of a patient's disease dictates it is not in the patient's best interest to wait for an unrelated marrow donor to be procured Have identified a back-up cell source in case of engraftment failure; the source can be autologous, related, or unrelated UNRELATED DONORS: Only G-CSF mobilized PBMC will be permitted as a HSC source on this protocol DONOR: Only G-CSF mobilized peripheral blood mononuclear cells (PBMC) only will be permitted as a hematopoietic stem cell (HSC) source on this protocol DONOR: Bone marrow is the preferred cell source Patients after allogeneic stem cell transplantation from a related or unrelated, HLA-matched or mismatched donor with the diagnosis of transplant related microangiopathy. Patients having received any of the following stem cell sources are eligible: G-CSF mobilized peripheral blood stem cells, bone marrow, umbilical cord blood. For Pre-allo Part A (before stem cell transplant): Partially matched donors (related or unrelated) and umbilical cord blood cells are excluded as the source of hematopoietic stem cells Back-up stem cell source Have identified a back-up cells source in case of engraftment failure; the source can be autologous, related or unrelated Any diagnosis, donor or source of hematopoietic stem cells (HSC) is allowed, including donor leukocyte infusions (DLI) Stem cell source: bone marrow, peripheral blood stem cell DONOR: Must be willing and able to undergo peripheral blood stem cell (PBSC) collection Scheduled to undergo a hematopoietic stem cell transplant for any cancer or non-cancer illness from any autologous, related or unrelated donor source including bone marrow, peripheral blood progenitor cell, or umbilical cord blood The patient must be undergoing allogeneic HCT from any donor (including matched related) with any stem cell source for any underlying condition 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 (100-140 mg/mg^2) Recipients of allogeneic hematopoietic cell transplantation (HCT) after either myeloablative or reduced intensity conditioning regimens; any donor source of stem cells is eligible Recipients of allogeneic hematopoietic cell transplantation (HCT) after either myeloablative or reduced intensity conditioning regimens; any donor source of stem cells is eligible Patients transplanted with hematopoietic stem cells from any source Peripheral blood stem cells must have been used as the stem cell source Donor must be willing to donate peripheral blood stem cells Peripheral blood stem cells, bone marrow, or umbilical cord blood may be used as the stem cell source Patients must have a related or unrelated peripheral blood stem cell donor as follows: