DONOR: Related donors with at least a haplotype at HLA-A, B, C, DR, and DQ loci that is shared with the recipient by high resolution typing, excluding an identical twin or unrelated donors matched 9-10/10 at HLA-A, B, C, DR, and DQ loci by high resolution typing HSCT Donor will be one of the following:\r\n* 5/6 or 6/6 (human leukocyte antigen [HLA]-A, B, DR) matched related donor\r\n* 7/8 or 8/8 (HLA-A, B, DR, C) matched unrelated donor. Matching in the unrelated setting must be at the allele level\r\n* Haploidentical related donor, defined as ? 3/6 (HLA-A, B, DR) matched\r\n* >= 4/6 (HLA-A, B, DR) umbilical cord blood (UCB). Matching in the UCB setting is at the antigen level. Recipients may receive either one or two UCB units. In the case of 2 UCB units, both units must have been at least 4/6 matched with the recipient DONOR: Arm A: All candidates for this study must have a human leukocyte antigen (HLA) (A, B, C, and DR) identical sibling who is willing to donate primed blood stem cells (preferred) or bone marrow, or have a 10/10 (A, B, C, DR and DQ) allele matched unrelated donor; DQ or DP mismatch is allowed per discretion of the principal investigator All candidates for this study must have a human leukocyte antigen (HLA) (A, B, C, DR) identical sibling who is willing to donate primed blood stem cells (preferred) or bone marrow, or have a 10/10 allele matched unrelated donor; all ABO blood group combinations of the donor/recipient are acceptable Participants who will undergo HCT from the following donor types are eligible:\r\n* 5/6 or 6/6 (HLA-A, B, DR) matched related donor\r\n* 7/8 or 8/8 (HLA-A, B, DR, C) matched unrelated donor. Matching in the unrelated setting must be at the allele level Availability of at least one 4/6 HLA-matched (HLA-A, B, and DR loci) cord blood unit from the National Marrow Donor Program (NMDP). The cord blood unit must contain a minimum total nucleated cell (TNC) (prior to thawing) of at least 2x10^7 cells per kilogram of recipient body weight All candidates for this study must have a human leukocyte antigen (HLA) (A, B, C, DR) identical siblings who is willing to donate bone marrow for primed blood stem cells or an 8/8 allele-matched unrelated donor Availability of an 8/8 matched donor at A, B, C, and DR loci; mismatch at human leukocyte antigen (HLA) DQ are permissible; matched related or unrelated donors are acceptable; peripheral blood or bone marrow stem cells are acceptable All candidates for this study must have a human leukocyte antigen (HLA) (A, B, C, DR) identical siblings who is willing to donate bone marrow or primed blood stem cells or a 10/10 allele matched unrelated donor; a single allele mismatch at A, B, C, DR or DQ and a killer immunoglobulin-like receptor (KIR) mismatch at C will be allowed; all ABO blood group combinations of the donor/recipient are acceptable Patient with no matched related donor who has a related haploidentical donor identified (=< 7/8 allele match at the A, B, C, DR loci) who is willing to undergo a bone marrow harvest and an NK cell collection approximately 2 weeks of the recipient's admission for transplant; the donor must be 16 years of age or older and weigh at least 110 pounds MATCHED RELATED DONOR: Related donor matched at 9/10 or 10/10 HLA-A, B, C, DR, and DQ loci by high resolution typing 63 MATCHED UNRELATED DONOR: Unrelated donor matched at 10/10 or 9/10 HLA-A, B, C, major histocompatibility complex, class II, DR beta 1 (DRB1), and major histocompatibility complex, class II, DQ beta 1 (DQB1) loci by high resolution typing Patient must have an identified HLA (A,B,C,DR) compatible related or unrelated donor who is age 16 years of age or older and weighs at least 110 pounds for the stem cell donation Must have matched unrelated donor (8 of 8 HLA match at A, B, C, and DR loci) by high resolution deoxyribonucleic acid (DNA) typing MATCHED RELATED DONOR: Related donors matched at HLA-A, B, C, DR, and DQ loci by high resolution typing (10/10 antigen/allele match) are acceptable donors; alternatively, a 9/10 matched related donor can be used MATCHED UNRELATED DONOR: Unrelated donor matched at 10/10 or 9/10 HLA-A, B, C, DR, and DQ loci by high resolution typing DONOR: HLA-genotypically or phenotypically 1 to 3 antigen mismatched (at the A, B, DR loci) related donors will be acceptable Recipient of mismatched (allele or antigen level) graft in more than one loci of HLAA, -B, -C or –DR loci will not be eligible, i.e. recipients of 2-antigen or 2-allelele mismatched graft Histocompatible stem cell donor: patients must have an human leukocyte antigen (HLA) matched related or unrelated donor (HLA A, B, C and DR) willing to donate for allogeneic hematopoietic transplantation; high resolution allele level typing is required for donors other than genotypically identical siblings MATCHED UNRELATED DONOR: Unrelated donor matched at HLA-A, B, C, and DR loci by high resolution typing (at 8/8 or 7/8 antigen/allele match) are acceptable donors Related or unrelated donor which is HLA-matched or mismatched in 1 HLA A, B, C, DR, or DQ locus is acceptable (i.e. >= 9/10 matched related or unrelated donor, matched with molecular high-resolution technique per current standard [std.] for bone marrow transplant [BMT] program); cord blood units must match patient at 4, 5, or 6/6 HLA class 1 serological & II molecular antigens with a minimum (min.) of 2 x 10^7 total number of nucleated cells (TNC)/kg recipient weight in the pre-thawed fraction; for patient lacking a matched related or unrelated donor or acceptable cord blood unit(s), a related haploidentical donor (=< 7/8 allele matched at A, B, C, DR loci) may be used DONOR: Volunteer unrelated donor matched at a minimum of seven of eight loci (HLA-A, B, C, DRB1), by high resolution typing (> 7/8 allele match) are acceptable donors \r\n* The preferred donor-recipient pair would be matched at all eight loci (8/8 allele match)\r\n* When an 8/8 allele-matched unrelated donor is not available, a single mismatch at HLA-A, -B, -C, or DRB1 will be acceptable in patients who meet all eligibility criteria and are 18-69 years of age\r\n** In the situation where more than one 7/8 match is available donor recipient pairs matched at HLA-DQ will be used Patients must have a related donor who is a two or more allele mismatch at the human leukocyte antigen (HLA)-A; B; C; DR and DQ loci; patients who have sibling donors with a one antigen mismatch due to recombination will not be enrolled in this protocol Patients of Dr. Gonzalez Medical clearance by Dr. Murphy All candidates for this study must have a human leukocyte antigen (HLA) (A, B, C, DR) identical siblings who is willing to donate bone marrow or primed blood stem cells or an 8/8 allele-matched unrelated donor 7 out of 8 at high resolution using deoxyribonucleic acid (DNA)-based typing with either antigen or allele mismatched HLA (-A, -B, -C, and -DR) or 8/8 HLA-mismatched with either double DQ mismatch (10/12) or combined DQ and DP mismatch Per Moffitt Cancer Center (MCC) bone marrow transplant (BMT) program practices, an allele level matched (8/8 human leukocyte antigen [HLA] A, B, C and DR) sibling or unrelated donor is preferred; if a matched donor is not found, mismatched unrelated or haploidentical donors may be considered