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 -DRB for potential haploidentical sibling donors is required. b)At least two potential umbilical cord blood units identified. Each unit must have a minimum of . x ^/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 . x ^/kg. Units must be HLA matched at a minimum of / to the recipient at HLA-A, HLA-B (at low resolution using DNA based typing) and HLA-DRB (at high resolution using DNA based typing). Confirmatory typing is not required for randomization. A sibling donor who is a / match at HLA-A and -B (intermediate or higher resolution) and -DRB (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 / match for HLA-A, -B, -C (at intermediate or higher resolution) and -DRB (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 / match at HLA-A, -B, -C, and -DRB (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 (/ match for human leukocyte antigen (HLA)-A and B at intermediate or high resolution and DRB at high resolution using DNA-based typing) or unrelated donor (/ match for HLA-A, B, C, and DRB 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 - weeks from referral and the medical need to choose a donor without further delay beyond that time. Cord blood unit(s) must be matched at a minimum of / to the recipient at HLA-A and B at low resolution using DNA-based typing and HLA-DRB at high resolution using DNA-based typing. Based on a published report from Eurocord, for single unit transplantation, the single unit pre-cryopreserved total nucleated cell (TNC) dose must be a minimum of . x^/kg recipient weight. For double cord transplants, each unit must have a minimum of . x^/kg pre-cryopreserved TNC and a minimum total of . x^/kg (sum of unit and unit ). For non-red blood cell depleted units, the minimum pre-cryopreserved TNC dose is . x^/kg recipient weight. or HLA haplo first degree relatives of the patient including biological parents, siblings or half siblings, or children with , , or mismatches using DNA-based typing. A unidirectional mismatch in either the graft versus host or host versus graft direction is considered a mismatch. The donor and recipient must be identical at a minimum of one allele (at high resolution DNA-based typing) at the following genetic loci: HLA-A, -B, -C, and DRB. DONOR: half match (haploidentical) at HLA-A, B, C, DRB based upon deoxyribonucleic acid (DNA) based typing methods; donors may be variably matched for the other allele (/ to /), but not fully matched 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 DRB 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 UCB units may be used to achieve the required cell dose\r\n* The UCB graft is matched at - human leukocyte antigen (HLA)-A, B, DRB antigens with the recipient; this may include - antigen mismatches at the A or B or DRB loci; unit selection based on cryopreserved nucleated cell dose and HLA-A,B, DRB using intermediate resolution A, B antigen and DRB allele typing\r\n* If UCB units are required to reach the target cell dose, each unit must be a - antigen match to the recipient DONOR: Must be a / human leukocyte antigen (HLA) genotypically match related or unrelated donor at all A, B, C, DRB, and DQB loci, as tested by deoxyribonucleic acid (DNA) analysis. CB units will be selected according to current Memorial Sloan Kettering Cancer Center (MSKCC) unit selection algorithm. High resolution allele HLA typing and recipient HLA antibody profile will be performed. Unit selection will occur based on HLA-match, total nucleated cell (TNC) and CD+ 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 / HLA-matched to the patient considering high-resolution -allele HLA typing.\r\n* Each CB unit will be required to have a cryopreserved TNC dose of at least . x ^ TNC/ recipient body weight (TNC/ kg).\r\n* Each CB unit will be required to have a cryopreserved CD+ cell dose of at least . x ^ CD+ cells/ recipient body weight (CD+/kg).\r\n* A minimum of one domestic will be reserved as a backup unit. Patients must be human leukocyte antigen (HLA) typed at high resolution using deoxyribonucleic acid (DNA) based typing at HLA-A, -B, -C and DRB and have an available related haploidentical bone marrow donor with , , or (out of ) HLA-mismatches; a unidirectional mismatch in either the graft versus host or host versus graft direction is considered a mismatch 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 / match for HLA-A and -B at intermediate (or higher) resolution, and HLA-DRB at high resolution using deoxyribonucleic acid (DNA)-based typing\r\n* Unrelated donor must be an / match at HLA-A, -B, -C and DRB at high resolution using DNA-based typing At least one haploidentical (/ antigen mismatched) related donor is available for bone marrow harvest\r\n* Molecular based HLA typing for the HLA-A, -B, -Cw, -DRB and -DQB loci to the resolution is needed to establish haploidentity\r\n* A minimum match of / is required\r\n* No availability of an / HLA-matched related or unrelated donor or clinical urgency for transplant (e.g., needed within - weeks) at which time an acceptable unrelated donor will not be available IMMUNE RECONSTITUTION STUDY ONLY: At least one haploidentical (/ antigen mismatched) related donor is available for bone marrow harvest\r\n* Molecular based HLA typing for the HLA-A, -B, -Cw, -DRB and -DQB loci to the resolution is needed to establish haploidentity\r\n* A minimum match of / is required\r\n* No availability of an / HLA-matched related or unrelated donor or clinical urgency for transplant (e.g., needed within - weeks) at which time an acceptable unrelated donor will not be available No suitable fully matched related (/ match for human leukocyte antigen [HLA]-A and B at intermediate or high resolution and DRB at high resolution using deoxyribonucleic acid [DNA] based typing) donor if under age years DONOR: The CB graft(s) must be matched at - HLA-A, B, DR Beta (DRB) loci with the recipient and therefore may include - mismatches at the A or B or DRB loci; unit selection will be based on cryopreserved nucleated cell dose and intermediate resolution A, B antigen and DRB 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 Cord blood units available through National Marrow Donor Program (NMDP) with the following minimal criteria;\r\n* HLA Match: / or better match (HLA A, B, DRB)\r\n* Cell dose: minimum of x ^ total number of nucleated cells (TNC)/kg pre thaw Molecular based human leukocyte antigen (HLA) typing will be performed for the HLA-A, -B, -Cw, DRB and DQB loci to the resolution adequate to establish haplo identity; a minimum match of / 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 - 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 / HLA match by high resolution typing Must have matched unrelated donor ( of HLA match at A, B, C, and DR loci) by high resolution deoxyribonucleic acid (DNA) typing Availability of a related or unrelated donor with a / or / match for HLA-A, B, C, and DRB. Related or unrelated umbilical cord blood unit with - antigen mismatch at human leukocyte antigen (HLA)-A and B (at low resolution) and DRB (at high resolution) with a total nucleated cell dose of >= x ^/kg Molecular based human leukocyte antigen (HLA) typing will be performed for the HLA-A, -B, -Cw, -DRB and -DQB loci to the resolution adequate to establish haplo-identity; a minimum match of / 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 - 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 / HLA-A, B, DRB loci with the recipient; therefore - mismatches at the A or B or DRB loci based on intermediate resolution A, B antigen and DRB 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 UCB units selected according to current Memorial Sloan-Kettering Cancer Center (MSKCC) unit selection algorithm; high resolution allele HLA typing will be performed; unit selection will occur based on allele HLA-match and cluster of differentiation (CD)+ dose DONOR: Cord blood (CB) donor selection will be based on institutional guidelines and in general should be selected to optimize both human leukocyte antigen (HLA) match and cell dose; additionally, CB grafts shall consist of one or two CB donors based on, but not exclusively determined by, cell dose (total nucleated cell [TNC]/kg and CD/kg), HLA matching and disease status and indication for transplant; attending preference will be allowed for single versus double unit as well as the degree of mismatching based on patient specific factors, as long as the following minimum criteria are met:\r\n* HLA matching\r\n** Minimum requirement: The CB graft(s) must be matched at a minimum at / HLA-A, B, DRB loci with the recipient. Therefore - mismatches at the A or B or DRB loci based on intermediate resolution A, B antigen and DRB 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 are met\r\n* Selection of two CB units is mandatory when a single cord blood unit does not meet the following criteria:\r\n** Match grade /; TNC Dose >= . x ^/kg\r\n** Match grade / or /; TNC dose >= . (+/- .) x ^/kg\r\n* If two CB units are used, the total cell dose of the combined units must be at least . x ^ TNC per kilogram recipient weight based on pre-cryopreservation numbers, with each CB unit containing a MINIMUM of . x ^ TNC/kg \r\n* The minimum recommended CD/kg cell dose should be x ^ CD/kg, total dose from a single or combined double\r\n* The unmanipulated CB unit(s) will be Food and Drug Administration (FDA) licensed or will be obtained under a separate investigational new drug (IND), such as the National Marrow Donor Program (NMDP) Protocol -CBA conducted under BB IND- or another IND sponsored by () a participating institution or () an investigator at FHCRC or one of the participating institutions\r\n* FHCRC only: Up to % of cord blood product, when ready for infusion, may be withheld for research purposes as long as thresholds for infused TNC dose are met; threshold for double unit transplantation is >= . x ^/kg; these products will be used to conduct studies involving the immunobiology of double cord transplantation and kinetics of engraftment If UCB units are required to reach the target cell dose, each unit must be a - HLAA, B, DRB antigen match to each other, as well as a - antigen match to the recipient DONOR: Minimum requirement: The cord blood (CB) unit must be matched at a minimum at / HLA-A, B antigens and DRB allele with the recipient; therefore, - mismatches at the A or B or DRB loci based on intermediate resolution at HLA-A, B and high resolution allele level typing at HLA- DRB are allowed Patients with a / or / related donor match are eligible or a -/ human leukocyte antigen (HLA)-A,B,C,DRB allele matched unrelated volunteer marrow and/or peripheral blood stem cell (PBSC) donor match The donor and recipient must have a human leukocyte antigen (HLA)-/ allelic match at the HLA-A, -B, -C, and DRB; 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 A preliminary search has identified:\r\n* An appropriate minimum / 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 x () total nucleated cell (TNC)/kg for a single umbilical cord blood (UCB) transplant AND\r\n* At least one potential / HLA-matched (HLA-A, -B, -C, and DRB) unrelated donor with a probability of % 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 years of age and at least kg, and a cord blood unit with at least . x () TNC/kg and a match of at least / by intermediate resolution for HLA-A and B and high resolution at DRB; 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 / or / HLA matched unrelated donor with high resolution matching at HLA-A, -B, -C, and DRB; DP matching or DP permissive should be achieved when possible using T-cell epitope strategy Available related donor that is at least an allele level haplotype-match at human leukocyte antigen (HLA)- A, B, C, DRB and DPB loci (DPB matching according to the permissive non-permissive dichotomy as stated by University of Wisconsin [UW] Histocompatibility Laboratory); a minimum match of / loci is required; an unrelated donor search is not required for a patient to be eligible for this protocol Related donor: must be an / match at HLA-A, -B, -C, (serologic typing or higher resolution) and -DRB (at high resolution using DNA based typing). A / related donor match is permitted only if an / unrelated donor cannot be identified. Unrelated donor: must be a / or / match at HLA-A, -B, -C, and -DRB (at high resolution using DNA based typing). DONOR: Volunteer unrelated donor matched at a minimum of seven of eight loci (HLA-A, B, C, DRB), by high resolution typing (> / allele match) are acceptable donors \r\n* The preferred donor-recipient pair would be matched at all eight loci (/ allele match)\r\n* When an / allele-matched unrelated donor is not available, a single mismatch at HLA-A, -B, -C, or DRB will be acceptable in patients who meet all eligibility criteria and are - years of age\r\n** In the situation where more than one / match is available donor recipient pairs matched at HLA-DQ will be used Patients must have a related, genotypically human leukocyte antigen (HLA) identical donor, or they must have a unrelated donor who is / HLA match by high resolution typing Patients must have an human leukocyte antigen (HLA)-compatible related or unrelated donor (one-antigen mismatched related donors are acceptable) willing to donate marrow or recombinant human-granulocyte colony-stimulating factor [rhG-CSF] mobilized peripheral blood stem cells; in the event of transplants from matched unrelated donors, a high-resolution allele match for HLA-A, -B, -C, -DRB (\ of match\) is required Patient must have an available unrelated donor with a / or / match for HLA-A, B, C, and DRB antigen; typing is by DNA techniques: intermediate resolution for A, B, and C, and high resolution for DRB; HLA-DQ typing is recommended but will not count in the match Patients donor must be a related or unrelated human leukocyte antigen (HLA) / allele-level match (HLA-A, B, C and DRB) Planned related or unrelated HCT, with HLA donor allele matching; related donor must be an / match for HLA-A, -B, and -C at intermediate (or higher) resolution, and -DRB at high resolution using deoxyribonucleic acid (DNA)-based typing; unrelated donor must be an / match at HLA-A, -B, -C, and -DRB 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) Planned related or unrelated HCT, with / (A, B, C, DRB) high/intermediate resolution HLA donor allele matching Patient must be the recipient of unrelated donor peripheral blood stem cell products; mismatches at both antigen and allele level will be eligible; match must be , , or out of loci (human leukocyte antigen [HLA] A, B, C and DRB) Each UCB unit must be matched at - human leukocyte antigen (HLA)-A, B, DRB antigens with the recipient; this may include - antigen mismatches at the A or B or DRB loci; each unit must be a - HLA-A, B, DRB antigen match to each other, not necessarily at the same loci they are matched to the recipient Related donor must be an / match for human leukocyte antigen (HLA)-A, -B, and -C at intermediate (or higher) resolution, and -DRB at high resolution using DNA-based typing. Pediatric related donors must weigh ? . kg., must have adequate peripheral venous catheter access for leukapheresis or must agree to placement of a central catheter, must be willing to () donate bone marrow and () 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 / match at HLA-A, -B, -C and -DRB 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 CD selection when cell processing begins within hours of the end of collection) RECIPIENT: Planned related HCT with / (A, B, C, DRB) high resolution HLA donor allele matching Patients must have a related or unrelated peripheral blood stem cell donor; sibling donor must be a / match for human leukocyte antigen (HLA)-A and -B at intermediate (or higher) resolution, and -DRB 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 / match at HLA-A, -B, -C and DRB 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 / match for HLA-A and -B at intermediate (or higher) resolution, and -DRB 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 / or / match at HLA-A, -B, -C and -DRB 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.