Patient should be human leukocyte antigen (HLA) typed (A, B, C, DR and DQ) during induction therapy phase or a written explanation for not undergoing HLA typing on the flow sheet Any human leukocyte antigen (HLA) type; (historic HLA typing is permitted) Human leukocyte antigen (HLA)-A, A, A, B, or B. No suitable human leukocyte antigen (HLA)-identical sibling donor DONOR: Human leukocyte antigen (HLA) haplo-identical matched related. Donor and recipient match each other for at least / human leukocyte antigen (HLA)-loci (HLA-A, B, C, and D-related [DR]). Lack human leukocyte antigen (HLA)-identical related donor Relapsed AML after human leukocyte antigen (HLA)-matched related or unrelated allogeneic hematopoietic cell transplant (per IWG definition of relapse) Availability of a CB unit matched with the patient at , , or / human leukocyte antigen (HLA) class I (serological) and II (molecular) antigens. Patients must have one related donor who is human leukocyte antigen (HLA) mismatched in the GVHD direction at two or more HLA loci Available human leukocyte antigen (HLA)-haploidentical donor that meets the criteria Human leukocyte antigen (HLA)-A* (HLA-A.) positivity by molecular subtyping Availability of appropriate, willing, human leukocyte antigen (HLA)-matched related stem cell donor No more than antigen mismatch at human leukocyte antigen (HLA)-A, B, C, DRB and DQB locus for either related or unrelated donor Human leukocyte antigen (HLA)-A* (HLA-A.) positivity by molecular subtyping (blood test or buccal swab, historical documentation acceptable) Patients must be human leukocyte antigen (HLA)-A+ Has a human leukocyte antigen (HLA)-matched or single allele-mismatched adult sibling serving as donor Human leukocyte antigen (HLA)-A positive based on flow cytometry DONOR: Human leukocyte antigen (HLA)-matched or antigen mismatched sibling donor Human leukocyte antigen (HLA)-identical sibling or / matched unrelated donor transplant No available suitable human leukocyte antigen (HLA)-matched donor Absence anti-human leukocyte antigen (HLA) antibodies specific for HLA class I antigens expressed by the coagulation factor III (thromboplastin, tissue factor) (F).cytosine deaminase (CD).carboxylesterase (CE) NSCs Patients must be human leukocyte antigen (HLA)-DP positive / human leukocyte antigen (HLA) matched family donor available Patients must be human leukocyte antigen (HLA)-A* positive Leukocyte >= ,/ul Patients must have a cord blood (CB) unit available which is matched with the patient at , , or / human leukocyte antigen (HLA) class I (serological) and II (molecular) antigens Human leukocyte antigen (HLA)-A* (HLA-A.) positivity by molecular subtyping Human leukocyte antigen (HLA)-A positive DONOR: Donor's high resolution human leukocyte antigen (HLA) typing must be available for review Patients must have a related donor who is human leukocyte antigen (HLA) mismatched at , , or antigens at the HLA-A; B; C; DR loci in the GVHD direction; (patients with related donors who are HLA identical or are a -antigen mismatch may be treated on this therapeutic approach, but will have their outcomes will not be part of the statistical aims of the study) Patients must have a related donor who is either human leukocyte antigen (HLA)-identical or a one antigen mismatch at the HLA- A; B; C; and DR loci Allogeneic transplant with a human leukocyte antigen (HLA)-identical sibling will be recommended for patients < years; if the patient refuses allogeneic transplant, they may still be eligible for this protocol 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 Patient has at least one medically fit first- or second-degree family member expected to be human leukocyte antigen (HLA) mismatched at -/ loci; in addition, the prospective donor is willing to voluntarily donate hematopoietic stem cells and sign consent forms Patients must have a cord blood unit available which is matched with the patient at , , or / human leukocyte antigen (HLA) class I (serological) and II (molecular) antigens ENROLLMENT: Confirmation that a cord blood donor which is matched with the recipient at a , , or / human leukocyte antigen (HLA) class I (serological) and HLA class II (molecular) antigens. Availability of a suitable human leukocyte antigen (HLA)-matched related donor Patients must have human leukocyte antigen (HLA)-A phenotype Human leukocyte antigen (HLA)-A, -A, -A, or -A positive Any human leukocyte antigen (HLA) type Patient has anti-human leukocyte antigen (HLA) antibodies specific for HLA antigens expressed by the HB.F.CD NSCs Patients must express human leukocyte antigen (HLA) -A+, -A+, or -A+ (% of patients) DONOR: Human leukocyte antigen (HLA)-identical related donors or Patients will be tested for human leukocyte antigen A (HLA-A) as determined by flow cytometry followed by molecular analysis of a peripheral blood specimen; however this result will not be an inclusion criterion Consenting first degree relative matched at / human leukocyte antigen (HLA) antigens (A, B, and DR) Lack of a suitable human leukocyte antigen (HLA)-matched related donor DONOR: Human leukocyte antigen (HLA) identical to recipient subject Patients must have a cord blood (CB) unit available which is matched with the patient at , , , or / human leukocyte antigen (HLA) class I (serological) and II (molecular) antigens Human leukocyte antigen (HLA)-A* (HLA-A.) positivity by molecular subtyping Human leukocyte antigen (HLA)-A positive Expression of human leukocyte antigen (HLA)-A: or HLA-A:. ELIGIBILITY FOR TREATMENT ON ARM : Patients must express human leukocyte antigen (HLA)-A* Human leukocyte antigen (HLA)-A-positive Human leukocyte antigen HLA A positive Expression of human leukocyte antigen (HLA)-A Patients must have one related donor who is human leukocyte antigen (HLA) mismatched in the GVHD direction at two or more HLA loci Patients must NOT have a human leukocyte antigen (HLA)-matched sibling Has a suitable human leukocyte antigen (HLA) haploidentical donor available Patients must have undergone a human leukocyte antigen (HLA) matched (sibling) allogeneic HCT for a hematologic or lymphoid malignancy other than chronic myelogenous leukemia (CML) who have recurrent or persistent disease and are otherwise eligible for donor leukocyte infusions; CML patients with persistent disease after receiving donor lymphocyte infusion of at least x ^ cells/kg will be eligible for CD+ memory T cell infusion A fully human leukocyte antigen (HLA) matched or single antigen/allele mismatched sibling or unrelated donor is available Human leukocyte antigen (HLA)-A positive by deoxyribonucleic acid (DNA) sequence analysis (by history or as part of this study); HLA testing can be done at local labs DONOR: Related or unrelated human leukocyte antigen (HLA) identical donors who are in good health and have no contra-indication to donation Donor is blood-related and human leukocyte antigen (HLA)-haploidentical to the recipient Patient must have a partially (>= / class I antigen) human leukocyte antigen (HLA)-matched (by serology or low resolution deoxyribonucleic acid [DNA] testing) relative able to serve as a donor DONOR: Donors will be selected to minimize human leukocyte antigen (HLA) mismatch in the host-versus-graft direction Patients must be human leukocyte antigen (HLA)-A* positive Available peptide-MHC pair that can be folded into a tetramer for which MCPyV TAg-specific cells can be generated and reactivity to cell lines expressing MCPyV TAg with the corresponding human leukocyte antigen (HLA) Human leukocyte antigen (HLA)-A*+ by serology by an ASHI accredited laboratory; Leukocyte count = / related or unrelated donors Planned related HCT with molecular / (haploidentical) intermediate/high resolution human leukocyte antigen (HLA) donor allele matching Availability of a willing and suitable human leukocyte antigen (HLA) identical related donor Recipient of -/ human leukocyte antigen (HLA)-matched (HLA-A, -B, -C, -DRB) allogeneic hematopoietic stem cell transplantation Human leukocyte antigen (HLA)-identical sibling donor Patients transplanted from related or unrelated, human leukocyte antigen (HLA)-matched or mismatched donors Participants must be human leukocyte antigen (HLA)-A positive Presence of human leukocyte antigen (HLA) antibodies No available human leukocyte antigen (HLA)-matched related donor Availability of an human leukocyte antigen (HLA) matched related donor Human leukocyte antigen (HLA) mismatched related or unrelated donor identified / or / Availability of human leukocyte antigen (HLA)-identical sibling donor The patient has a human leukocyte antigen (HLA)-matched donor and is eligible for allogeneic transplantation for CML treatment.