Human leukocyte antigen (HLA) 8/8 or 7/8 matched related or unrelated donor available, as determined by antigen or allele level typing at HLA A, B, C, and allele level typing at major histocompatibility complex, class II, DR beta 1 (HLA DRB1) DONOR: HLA-matching must be based on results of high resolution typing at HLA-A, –B, -C, -DRB1, and –DQB HLA typing prior to referral (consultation with HCT physician). If a subject has had HLA typing with accompanying documentation that full siblings were not HLA typed and that a search of the unrelated donor registry was not performed the subject will be considered eligible. Documentation will be reviewed and adjudicated by the Protocol Officer or his/her designee. The donor and recipient must have an HLA-8/8 allelic match at the HLA-A, -B, -C, and - DRB1 loci. High-resolution typing is required for all alleles. Only matched unrelated donors are acceptable for this trial. DONOR: Sibling of any patient enrolled on this protocol proven by ABO typing, human leukocyte antigen (HLA) typing and variable number tandem repeat (VNTR) analysis to be syngeneic with the patient (e.g. identical twin) Patients must have available both: a)One or more potential related mismatched donors (biologic parent(s) or siblings (full or half) or children). At least low resolution DNA based human leukocyte antigen (HLA) typing at HLA-A, -B, and -DRB1 for potential haploidentical sibling donors is required. b)At least two potential umbilical cord blood units identified. Each unit must have a minimum of 1.5 x 10^7/kg pre-cryopreserved total nucleated cell dose. For non-red blood cell depleted units, the minimum pre-cryopreserved total nucleated cell dose of each unit must be at least 2.0 x 10^7/kg. Units must be HLA matched at a minimum of 4/6 to the recipient at HLA-A, HLA-B (at low resolution using DNA based typing) and HLA-DRB1 (at high resolution using DNA based typing). Confirmatory typing is not required for randomization. Units must be HLA matched at a minimum of 4/6 to the recipient at HLA -A, HLA-B (at low resolution using DNA based typing), and HLA -DRB1 (at high resolution using DNA based typing). DONOR: HLA matched unrelated donor according to Standard Practice HLA matching criteria:\r\n* Matched HLA-A, -B, -C, and -DRB1 alleles by high resolution typing\r\n* Only a single allele disparity will be allowed for HLA-A, B, or C as defined by high resolution typing HLA-matched unrelated donors (HLA-A, B, C, and DRB1 matched based on high-resolution typing) capable and willing to donate PBSC (Arms C and D) Available HLA 3-5/6 matched genotypically haploidentical family member donor (based upon A, B intermediate and DRB1 high resolution HLA typing; additional C and DQB1 typing may be necessary to accurately assign haplotypes) DONOR: Human leukocyte antigens (HLA)-identical related donors or unrelated donors matched for HLA-A, B, C, DRB1, and DQB1 or mismatched for a single allele at HLA-A, B, C, DRB1 or a single DQB1 antigen or allele mismatch by high resolution deoxyribonucleic acid (DNA) typing DONOR: Unrelated Umbilical Cord Blood: Unit selection is based on the cryopreserved total nucleated cell (TNC) dose and matching at HLA-A, B antigen level and DRB1 allele level typing; while HLA-C antigen/allele level typing is not considered in the matching criteria, if available, may be used to optimize unit selection GRAFT CRITERIA: \r\n* UCB units will be selected according to current umbilical cord blood graft selection algorithm; one or 2 UCB units may be used to achieve the required cell dose\r\n* The UCB graft is matched at 4-6 human leukocyte antigen (HLA)-A, B, DRB1 antigens with the recipient; this may include 0-2 antigen mismatches at the A or B or DRB1 loci; unit selection based on cryopreserved nucleated cell dose and HLA-A,B, DRB1 using intermediate resolution A, B antigen and DRB1 allele typing\r\n* If 2 UCB units are required to reach the target cell dose, each unit must be a 4-6 antigen match to the recipient 2 CB units will be selected according to current Memorial Sloan Kettering Cancer Center (MSKCC) unit selection algorithm. High resolution 8 allele HLA typing and recipient HLA antibody profile will be performed. Unit selection will occur based on HLA-match, total nucleated cell (TNC) and CD34+ cell dose adjusted per patient body weight. The bank of origin will also be taken into account. Donor specific HLA antibodies, if present, will also be taken into consideration and may influence the selection of the graft.\r\n* Each CB unit must be at least 3/8 HLA-matched to the patient considering high-resolution 8-allele HLA typing.\r\n* Each CB unit will be required to have a cryopreserved TNC dose of at least 1.5 x 10^7 TNC/ recipient body weight (TNC/ kg).\r\n* Each CB unit will be required to have a cryopreserved CD34+ cell dose of at least 1.0 x 10^5 CD34+ cells/ recipient body weight (CD34+/kg).\r\n* A minimum of one domestic will be reserved as a backup unit. HLA-MATCHED UNRELATED DONOR: FHCRC matching allowed will be grades 1.0 to 2.1; unrelated donors who are prospectively:\r\n* Matched for HLA-A, B, C, DRB1 and DQB1 by high resolution typing; \r\n* Only a single allele disparity will be allowed for HLA-A, B, or C as defined by high resolution typing HLA-MISMATCHED UNRELATED DONOR: HLA-matching must be based on results of high resolution typing at HLA-A, –B, -C, - DRB1, and –DQ At least one haploidentical (5/10 antigen mismatched) related donor is available for bone marrow harvest\r\n* Molecular based HLA typing for the HLA-A, -B, -Cw, -DRB1 and -DQB1 loci to the resolution is needed to establish haploidentity\r\n* A minimum match of 5/10 is required\r\n* No availability of an 8/8 HLA-matched related or unrelated donor or clinical urgency for transplant (e.g., needed within 4-8 weeks) at which time an acceptable unrelated donor will not be available IMMUNE RECONSTITUTION STUDY ONLY: At least one haploidentical (5/10 antigen mismatched) related donor is available for bone marrow harvest\r\n* Molecular based HLA typing for the HLA-A, -B, -Cw, -DRB1 and -DQB1 loci to the resolution is needed to establish haploidentity\r\n* A minimum match of 5/10 is required\r\n* No availability of an 8/8 HLA-matched related or unrelated donor or clinical urgency for transplant (e.g., needed within 4-8 weeks) at which time an acceptable unrelated donor will not be available Patient must be HLA typed at high resolution using deoxyribonucleic acid (DNA) based typing at the following loci: HLA-A, -B, -C, and DRB1 Patients must be major histocompatibility complex, class I, A (HLA-A)*02 by low resolution typing, and HLA-A*02:01 by one of the high resolution type results An 10/10 or 8/8 human leukocyte antigen (HLA) matched (high resolution typing at A, B, C, DRB1, DQ1) sibling or unrelated donor DONOR: The CB graft(s) must be matched at 4-6 HLA-A, B, DR Beta 1 (DRB1) loci with the recipient and therefore may include 0-2 mismatches at the A or B or DRB1 loci; unit selection will be based on cryopreserved nucleated cell dose and intermediate resolution A, B antigen and DRB1 allele typing for determination of HLA-match; while HLA-C antigen/allele level typing is not considered in the matching criteria, if available, it may be used to optimize unit selection DONOR: HLA typing per University of Alabama (UAB) standard Molecular based human leukocyte antigen (HLA) typing will be performed for the HLA-A, -B, -Cw, DRB1 and –DQB1 loci to the resolution adequate to establish haplo identity; a minimum match of 5/10 is required; an unrelated donor search is not required for a patient to be eligible for this protocol if the clinical situation dictates an urgent transplant; clinical urgency is defined as 6-8 weeks from referral or low-likelihood of finding a matched, unrelated donor Patients must have a related, genotypically HLA identical donor, or they must have a unrelated donor who is 8/8 HLA match by high resolution typing An 8/8 human leukocyte antigen (HLA) matched (high resolution typing at A, B, C, DRB1) sibling or unrelated donor DONOR: Donor's high resolution human leukocyte antigen (HLA) typing must be available for review Related or unrelated umbilical cord blood unit with 0-1 antigen mismatch at human leukocyte antigen (HLA)-A and B (at low resolution) and DRB1 (at high resolution) with a total nucleated cell dose of >= 4 x 10^7/kg DONOR: Any matched sibling donor (matched at HLA A, B, C by intermediate resolution typing and HLA-DRB1 by high resolution typing), or unmatched unrelated donor (matched at HLA A, B, C, DRB1 by high resolution typing) will be considered a suitable donor Molecular based human leukocyte antigen (HLA) typing will be performed for the HLA-A, -B, -Cw, -DRB1 and -DQB1 loci to the resolution adequate to establish haplo-identity; a minimum match of 5/10 is required; an unrelated donor search is not required for a patient to be eligible for this protocol if the clinical situation dictates an urgent transplant; clinical urgency is defined as 6-8 weeks from referral or low-likelihood of finding a matched, unrelated donor DONOR: Human leukocyte antigen (HLA) matching:\r\n* Minimum requirement: The cord blood (CB) graft(s) must be matched at a minimum at 4/6 HLA-A, B, DRB1 loci with the recipient; therefore 0-2 mismatches at the A or B or DRB1 loci based on intermediate resolution A, B antigen and DRB1 allele typing for determination of HLA-match is allowed\r\n* HLA-matching determined by high resolution typing is allowed per institutional guidelines as long as the minimum criteria (above) are met 2 UCB units selected according to current Memorial Sloan-Kettering Cancer Center (MSKCC) unit selection algorithm; high resolution 8 allele HLA typing will be performed; unit selection will occur based on 8 allele HLA-match and cluster of differentiation (CD)34+ dose Patients must have an HLA-A, B, DRB1 identical sibling donor; patients and donors will be typed for HLA-A and B using serological or molecular techniques and for DRB1 using high resolution molecular typing DONOR: all matching will be performed at the allele level, with high resolution typing Must be HLA compatible in 10/10 or 9/10 alleles by 4 digit/allele high-resolution molecular genotyping Patients must have an =< 1 antigen mismatched HLA-A, B, DRB 1 unrelated donor or =< 1 antigen mismatched related (non-HLA-matched sibling) or =< 2 antigen mismatched unrelated umbilical cord blood (UCB) donor; patients and donors will be typed for HLA-A and B using intermediate or high resolution molecular techniques and for DRB 1 using high resolution molecular typing The unrelated cord blood donor(s) must be 4-6/6 human leukocyte antigen (HLA)-A, B, DRB1 matched with the recipient (HLA matching using molecular techniques: A and B to antigen level resolution and DR to allele level resolution) HLA-Identical Sibling (6/6): Minimal typing necessary is serologic typing for class I (AB) and molecular typing for class II (DRB1) Matched Unrelated Donor (8/8): Molecular identity at HLA A, B, C and DRB1 by high-resolution typing Matched Related and Unrelated Donor (7/8): high-resolution molecular typing at the following loci is required: HLA A, B, C and DRB1 DONOR: Minimum requirement: The cord blood (CB) unit must be matched at a minimum at 4/6 HLA-A, B antigens and DRB1 allele with the recipient; therefore, 0-2 mismatches at the A or B or DRB1 loci based on intermediate resolution at HLA-A, B and high resolution allele level typing at HLA- DRB1 are allowed HLA typing should be performed at registration, if possible An eligible HLA-identical donor (either related or unrelated) should be available; in sibling donors, low resolution HLA typing (A,B,DR) will be considered sufficient; in the case of unrelated donors, high-resolution class I and II typing (A, B, C, DRB1 and DQ) should be matched at all 10 loci; donors must be willing and able to undergo peripheral blood progenitor mobilization \r\n* HLA-identical sibling (6/6): the donor must be determined to be an HLA-identical sibling (6/6) by serologic typing for class (A, B) and low resolution molecular typing for class II (DRB1)\r\n* Matched unrelated donor (10/10): high resolution molecular typing at the following loci is required: HLA-A, -B, -C, -DRBL, and –DQB1\r\n* NOTE: for matched donors – will allow select 1 antigen mismatched sibling donors and unrelated donors in accordance with site institutional standard, as long as matched at HLA-A, HLA-B, HLA-C, and DRB1, and with advanced discussion/approval by the Study Chair and the bone marrow transplant (BMT) co-chair Availability of a 6/6 human leukocyte antigen (HLA) matched sibling defined by class I (HLA –A and B) serologic typing (or higher resolution) and class II (HLA-DRB1) molecular typing DONORS will be a HLA-identical sibling or HLA-matched unrelated donor; unrelated donors are required to be matched by high resolution allele level typing for HLA-A, B, C and DRB1 and intermediate resolution Sequence Specific Oligonucleotide Probes (SSOP), identifying alleles in groups of related families historically defined as antigens for DQB1; an unrelated donor is considered matched if patient and donor share HLA-A, B, C alleles with identical sequences at exons 2 and 3, DRB1 alleles with identical sequences at exon 2, and DQB1 results that include the same allele groups An 8/8 or 7/8 human leukocyte antigen (HLA)–matched non-syngeneic donor is available and eligible to donate hematopoietic stem cells following institutional guidelines for bone marrow transplant (BMT) procedure; high-resolution HLA typing is required at HLA-A, -B, -C and -DR alleles DONOR: Unrelated donors matched for HLA-A, B, C, DRB1, and DQB1 as defined by high resolution deoxyribonucleic acid (DNA) typing; mismatch for one HLA allele is allowed The donor and recipient must have a human leukocyte antigen (HLA)-8/8 allelic match at the HLA-A, -B, -C, and –DRB1; high-resolution typing is required for all alleles; donors will be identified according to the institutional bone marrow transplant (BMT) program clinical practice guidelines, which is available to all University of Michigan BMT protocol team members at the internal website Donors: Related donor who is human leukocyte antigen (HLA) genotypically identical at least at one haplotype and may be genotypically or phenotypically identical for serological typing for HLA-A, B, -C, and at the allele level for -DRB1 and -DQB1; related donors must be a match or a single allele mismatch at HLA-A, B, and C (at highest resolution available at the time of donor selection) and matched at DRB1 and DQB1 by deoxyribonucleic acid (DNA) typing Donors: Unrelated donors who are prospectively:\r\n* Matched for HLA-A, B, C, DRB1 and DQB1 by DNA typing at the highest resolution routinely available at the time of donor selection\r\n* Only a single allele disparity will be allowed for HLA-A, B, or C as defined by high resolution typing (no mismatching for DRB1 or DQB1 is allowed) Lack of suitable conventional donor (HLA identical sibling or HLA phenotypically identical relative or 10/10 unrelated donor evaluated using high resolution molecular typing) or presence of rapidly progressive disease not permitting time to identify an unrelated donor Unrelated Donor: FHCRC matching allowed will be grades 1.0 to 2.1: Unrelated donors who are prospectively:\r\n1) Matched for HLA-A, B, C, DRB1 and DQB1 by high resolution typing;\r\n2) Only a single allele disparity will be allowed for HLA-A, B, or C as defined by high resolution typing A preliminary search has identified:\r\n* An appropriate minimum 4/6 matched umbilical cord unit at intermediate resolution at human leukocyte antigen (HLA)-A and B, and high resolution at HLA-DRB with a cell dose above 1 x 10(7) total nucleated cell (TNC)/kg for a single umbilical cord blood (UCB) transplant AND\r\n* At least one potential 8/8 HLA-matched (HLA-A, -B, -C, and –DRB1) unrelated donor with a probability of 70% AND\r\n* Availability of a potential related haploidentical donor An adequate graft for the defined donor type\r\n* Haplo-cord requires a haploidentical adult donor of 14 years of age and at least 50 kg, and a cord blood unit with at least 1.0 x 10(7) TNC/kg and a match of at least 4/6 by intermediate resolution for HLA-A and B and high resolution at DRB1; donor provides standard of care consent for harvest following institutional policy; any donor samples or donor research data would be obtained on separate donor research protocol\r\n* For MUD requires a 7/8 or 8/8 HLA matched unrelated donor with high resolution matching at HLA-A, -B, -C, and DRB1; DP matching or DP permissive should be achieved when possible using T-cell epitope strategy DONOR: FHCRC matching allowed will be grade 1.0 to 2.1: unrelated donors who are prospectively: matched for HLA-A, B, C, DRB1 and DQB1 by high resolution typing; only a single allele disparity will be allowed for HLA-A, B, or C as defined by high resolution typing DONOR: HLA-matched unrelated donors (HLA-A, B, C, and DRB1 matched based on high-resolution typing) capable and willing to donate PBSC DONOR: FHCRC matching allowed will be grades 1.0 to 2.1: Unrelated donors who are prospectively:\r\n* Matched for human leukocyte antigen (HLA)-A, B, C, DRB1 and DQB1 by high resolution typing\r\n* Only a single allele disparity will be allowed for HLA-A, B, or C as defined by high resolution typing UNRELATED DONORS: \r\n* Fred Hutchinson Cancer Research Center (FHCRC) matching allowed will be Grades 1.0 to 2.1; Unrelated donors who are prospectively: \r\n* Matched for HLA-A, B, C, DRB1 and DQB1 by high resolution typing\r\n* Only a single allele disparity will be allowed for HLA-A, B, or C as defined by high resolution typing Patients having received a preceding allogeneic transplantation from either a human leukocyte antigen (HLA)-matched related or unrelated donor are eligible for this protocol\r\n* Related donor: HLA genotypically identical at least at one haplotype and may be phenotypically or genotypically identical at the allele level at HLA A, B, C, DRB1, and DQB1\r\n* Unrelated donor who are prospectively: \r\n** Matched for HLA-A, B, C, DRB1 and DQB1 by high resolution typing; OR\r\n** Only a single allele disparity will be allowed for HLA-A, B, or C as defined by high resolution typing DONOR: Fred Hutchinson Cancer Research Center (FHCRC) matching allowed will be grades 1.0 to 2.1; unrelated donors who are prospectively: matched for HLA-A, B, C, DRB1 and DQB1 by high resolution typing; only a single allele disparity will be allowed for HLA-A, B, or C as defined by high resolution typing DONOR: HLA genotypically identical sibling or unrelated donor; unrelated donors are required to be matched by standard molecular methods at the intermediate resolution level at HLA-A, B, C and DRB1 and the allele level at DQB1 A 10/10 HLA matched (high resolution typing at A, B, C, DRB1, DQ1) sibling or unrelated donor. Patient must qualify with a study specific HLA typing assay. Patients must have a related, genotypically human leukocyte antigen (HLA) identical donor, or they must have a unrelated donor who is 8/8 HLA match by high resolution typing Planned related HCT with molecular 3/6 (haploidentical) intermediate/high resolution human leukocyte antigen (HLA) donor allele matching Planned related or unrelated HCT, with 8/8 (A, B, C, DRB1) high/intermediate resolution HLA donor allele matching Patients must have received transplantation from donors (both related and unrelated) who are identical at 8 HLA loci (A, B, C and DRbeta1), or mismatched at no more than 1 locus (7/8); among related donors, HLA C typing is not required (6/6 HLA matches); class I typing is to be performed by polymerase chain reaction (PCR)-sequence specific primers (SSP) techniques and complement-dependent cytotoxicity (CDC) techniques; class II typing is performed by PCR-restriction fragment length polymorphism (RFLP) +/- PCR-SSP techniques RECIPIENT: Planned related HCT with 6/6 (A, B, C, DRB1) high resolution HLA donor allele matching Human leukocyte antigen (HLA) matching:\r\n* HLA typing will be performed by high resolution molecular deoxyribonucleic acid (DNA) typing for HLA class 1 A, B and C, and class II DRB1 and DQB1 alleles DONOR: HLA typing will be performed by high resolution molecular DNA typing for HLA class 1 A, B and C, and class II DRB1 and DQB1 alleles DONOR: unrelated adult donors must be matched by high resolution molecular DNA typing at 7/8 or 8/8 HLA A, B and C (class I) and HLA DRB1 (class II) alleles, or 9/10 or 10/10 HLA A, B and C (class I) and HLA DRB1 and DQB1(class II) alleles DONOR: related donors must be matched by high resolution molecular DNA typing at 5/10, 6/10, 7/10, 8/10, 9/10 or 10/10 HLA A, B and C (class I) and HLA class II DRB1 and DQB1 alleles Availability of a related haploidentical donor with ? 4/8 but < 7/8, or ? 5/10 but < 9/10 matches at the HLA-A, -B, -C, -DRB1, and/or -DQB1 loci, as determined by high resolution human leukocyte antigen (HLA)-typing