A sibling donor who is a 6/6 match at HLA-A and -B (intermediate or higher resolution) and -DRB1 (at high resolution using DNA-based typing) and must be willing to donate peripheral blood stem cells and meet institutional criteria for donation OR A related donor (other than sibling) who is a 8/8 match for HLA-A, -B, -C (at intermediate or higher resolution) and -DRB1 (at high resolution using DNA-based typing) and must be willing to donate peripheral blood stem cells and meet institutional criteria for donation OR An unrelated donor who is an 8/8 match at HLA-A, -B, -C, and -DRB1 (at high resolution using DNA-based typing) and must be willing to donate peripheral blood stem cells and meet institutional criteria for donation. No suitable fully matched related (6/6 match for human leukocyte antigen (HLA)-A and B at intermediate or high resolution and DRB1 at high resolution using DNA-based typing) or unrelated donor (8/8 match for HLA-A, B, C, and DRB1 at high resolution using DNA-based typing) available. Search for an unrelated donor and enrollment on this protocol may be abandoned if the clinical situation dictates an urgent transplant in the best medical judgment of the treating provider. The definition of clinical urgency may include a low likelihood of identifying a suitable matched unrelated donor within 6-8 weeks from referral and the medical need to choose a donor without further delay beyond that time. Patients undergoing an unmodified transplant must have a related or unrelated marrow or peripheral blood stem cell (PBSC) donor as follows:\r\n* Sibling donor must be a 6/6 match for HLA-A and -B at intermediate (or higher) resolution, and HLA-DRB1 at high resolution using deoxyribonucleic acid (DNA)-based typing\r\n* Unrelated donor must be an 8/8 match at HLA-A, -B, -C and –DRB1 at high resolution using DNA-based typing No suitable fully matched related (6/6 match for human leukocyte antigen [HLA]-A and B at intermediate or high resolution and DRB1 at high resolution using deoxyribonucleic acid [DNA] based typing) donor if under age 25 years Related donor: must be an 8/8 match at HLA-A, -B, -C, (serologic typing or higher resolution) and -DRB1 (at high resolution using DNA based typing). A 7/8 related donor match is permitted only if an 8/8 unrelated donor cannot be identified. Unrelated donor: must be a 7/8 or 8/8 match at HLA-A, -B, -C, and -DRB1 (at high resolution using DNA based typing). Patient must have an available unrelated donor with a 7/8 or 8/8 match for HLA-A, B, C, and DRB1 antigen; typing is by DNA techniques: intermediate resolution for A, B, and C, and high resolution for DRB1; HLA-DQ typing is recommended but will not count in the match Planned related or unrelated HCT, with HLA donor allele matching; related donor must be an 8/8 match for HLA-A, -B, and -C at intermediate (or higher) resolution, and -DRB1 at high resolution using deoxyribonucleic acid (DNA)-based typing; unrelated donor must be an 8/8 match at HLA-A, -B, -C, and -DRB1 at high resolution using DNA-based typing; patients undergoing a second allogeneic (allo) HCT are not eligible (patients who have undergone a previous autologous HCT are eligible) Related donor must be an 8/8 match for human leukocyte antigen (HLA)-A, -B, and -C at intermediate (or higher) resolution, and -DRB1 at high resolution using DNA-based typing. Pediatric related donors must weigh ? 25.0 kg., must have adequate peripheral venous catheter access for leukapheresis or must agree to placement of a central catheter, must be willing to (1) donate bone marrow and (2) receive G-CSF followed by donation of peripheral blood stem cells (product to be determined by randomization post enrollment) and must meet institutional criteria for donation. Unrelated donor must be an 8/8 match at HLA-A, -B, -C and -DRB1 at high resolution using DNA-based typing. Unrelated donor must be medically eligible to donate according to National Marrow Donor Program (NMDP) (or equivalent donor search organization) criteria. At time of enrollment, the donor should not have any known preferences or contraindications to donate bone marrow or peripheral blood stem cells. (Selection of unrelated donors is to be performed according to institutional practice. It is recommended that the time from collection to initiation of the cell processing be considered when prioritizing donors, as data shows better results for CD34 selection when cell processing begins within 36 hours of the end of collection) Patients must have a related or unrelated peripheral blood stem cell donor; sibling donor must be a 6/6 match for human leukocyte antigen (HLA)-A and -B at intermediate (or higher) resolution, and -DRB1 at high resolution using deoxyribonucleic acid (DNA)-based typing, and must be willing to donate peripheral blood stem cells and meet institutional criteria for donation; unrelated donor must be 8/8 match at HLA-A, -B, -C and –DRB1 at high resolution using DNA-based typing; unrelated donor must be willing to donate peripheral blood stem cells and be medically eligible to donate stem cells according to National Marrow Donor Program (NMDP) criteria Sibling donor must be a 6/6 match for HLA-A and -B at intermediate (or higher) resolution, and -DRB1 at high resolution using DNA-based typing, and must be willing to donate peripheral blood stem cells and meet institutional criteria for donation. Unrelated donor must be a 7/8 or 8/8 match at HLA-A, -B, -C and -DRB1 at high resolution using DNA-based typing. Unrelated donor must be willing to donate peripheral blood stem cells and be medically cleared to donate stem cells according to National Marrow Donor Program (NMDP) criteria.