Availability of appropriate HLA partially-matched and restricted tabelecleucel cell product HLA-matched or partially matched (7/8 or 8/8) related or unrelated donor is available to donate Must have two partially HLA-matched CBUs for part 1; and one partially HLA-matched CBU for part 2 Patients developing EBV lymphomas or lymphoproliferative disorders following an allogeneic organ transplant; in these cases, the lymphoma is usually of recipient origin; EBV-specific T-cells will be selected from the MSKCC bank expanded from an EBV-seropositive normal donor who is at least matched for 1) 2 HLA antigens and 2) one restricted allele with the EBV lymphoma; if the origin of the lymphoma is unknown, T-cells partially matched with the recipient transplant will be used, since these lymphomas are usually of host origin; using this approach to donor selection, it is expected that the EBV-specific, HLA restricted cytotoxic T-cells expanded from the HLA partially-matched donor would be able to recognize and kill lymphoma cells presenting EBV antigens in the context of an appropriate HLA restricting element; priority will be given to the use of partially matched EBV specific T cells known to be restricted by an HLA allele shared by the lymphoma (or, if known, the patient) Partially matched related donors will be at least haploidentical (matched at >= 3 of 6 HLA A, B, DRB1 loci) Matched sibling donor (HLA 8/8), if available, or a unrelated partially HLA matched single unit based on the following priority:\r\n* First (1st) priority: 4/6 matched unit, cell dose >= 5 x 10^7 nucleated cells/kg\r\n* Second (2nd) priority: 5/6 matched unit, cell dose >= 4 x 10^7 nucleated cells/kg\r\n* Third (3rd) priority: 6/6 matched unit, cell dose >= 3 x 10^7 nucleated cells/kg DONOR: Donor must be related to patient and be partially (>= 3/6 antigen) HLA-matched Patients must have a matched or partially matched UCB unit with >= 2.5 x10^7 nucleated cells/kg of recipient weight at the time of cryopreservation Patients must have two partially HLA-matched CBUs DONOR: Partially HLA matched family member DONOR: donor must be related to patient and be partially (>= 3/6 antigen) HLA-matched Patients must have one or two partially HLA-matched CBUs DONOR: Partially HLA matched family member Has a suitable partially human leukocyte antigen (HLA)-matched (>= 3 of 6) family member donor Availability of appropriate partially HLA-matched and restricted tabelecleucel cell product Has a suitable human leukocyte antigen (HLA) partially matched family member donor (HLA identical or phenotypically matched 8/8 family member donors will not be used on this protocol) Eligible patients will have one of the following sources of donor stem cells:\r\n* Human leukocyte antigen (HLA) matched family member\r\n* Partially matched family member (mismatched for a single HLA locus at A, B, C or DR)\r\n* Fully HLA matched or partially mismatched unrelated marrow or peripheral blood stem cells (per institutional donor selection standards)\r\n* HLA matched or partially mismatched (at least 4/6 match at A, B, DR) cord blood Eligible participants will only be transplanted for standard clinical indications, which include hematologic malignancies such as acute leukemia, high risk lymphoma, and multiple myeloma; much less commonly there are non-malignant indications for transplants; these include aplastic anemia and severe hemoglobinopathies; no patients will be transplanted for the primary purpose of HIV eradication; patients will undergo one of the following types of transplant:\r\n* Myeloablative, human leukocyte antigen (HLA) matched or partially HLA-mismatched (haploidentical), alloHSCT that includes high-dose post-transplantation cyclophosphamide (Cy)\r\n* Nonmyeloablative, HLA matched or partially HLA-mismatched, alloHSCT that includes high-dose post-transplantation Cy RELATED DONOR: a 5/10, 6/10, 7/10, 8/10, 9/10 or 10/10 matched (or partially matched) family donor will be required for study entry; HLA typing can be performed at any time prior to conditioning and must be performed twice on both donor and recipient Has a suitable single haplotype matched (>= 3 of 6) family member donor DONOR: At least single haplotype matched (>= 3 of 6) family member Has a suitable single haplotype matched (>= 3 of 6) family member donor DONOR: At least single haplotype matched (>= 3 of 6) family member Subject is a family member or employee of the investigator Has a suitable single haplotype matched (>= 3 of 6) family member donor DONOR: At least single haplotype matched (>= 3 of 6) family member At least single haplotype matched (>= 3 of 6) family member DONOR: At least single haplotype matched (>= 3 of 6) family member 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)-A1, A2, A3, B35, or B51. No suitable human leukocyte antigen (HLA)-identical sibling donor Donor and recipient match each other for at least 7/8 human leukocyte antigen (HLA)-loci (HLA-A, B, C, and D-related [DR]). Lack human leukocyte antigen (HLA)-identical related donor 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*0201 (HLA-A2.1) positivity by molecular subtyping Human leukocyte antigen (HLA)-A*0201 (HLA-A2.1) positivity by molecular subtyping (blood test or buccal swab, historical documentation acceptable) Patients must be human leukocyte antigen (HLA)-A2+ Human leukocyte antigen (HLA)-A2 positive based on flow cytometry Absence anti-human leukocyte antigen (HLA) antibodies specific for HLA class I antigens expressed by the coagulation factor III (thromboplastin, tissue factor) (F3).cytosine deaminase (CD).carboxylesterase (CE) NSCs Patients must be human leukocyte antigen (HLA)-DP4 positive Patients must be human leukocyte antigen (HLA)-A*0201 positive Leukocyte >= 3,000/ul Human leukocyte antigen (HLA)-A*0201 (HLA-A2.1) positivity by molecular subtyping Human leukocyte antigen (HLA)-A2 positive 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 Patients must have human leukocyte antigen (HLA)-A2 phenotype Human leukocyte antigen (HLA)-A1, -A2, -A3, or -A31 positive Any human leukocyte antigen (HLA) type Patient has anti-human leukocyte antigen (HLA) antibodies specific for HLA antigens expressed by the HB1.F3.CD NSCs Patients must express human leukocyte antigen (HLA) -A1+, -A2+, or -A3+ (80% of patients) DONOR: Human leukocyte antigen (HLA)-identical related donors or DONOR: Human leukocyte antigen (HLA) identical to recipient subject Human leukocyte antigen (HLA)-A*0201 (HLA-A2.1) positivity by molecular subtyping Human leukocyte antigen (HLA)-A2 positive Expression of human leukocyte antigen (HLA)-A:0201 or HLA-A:2402. ELIGIBILITY FOR TREATMENT ON ARM 1: Patients must express human leukocyte antigen (HLA)-A*0201 Human leukocyte antigen (HLA)-A2-positive Human leukocyte antigen HLA A2 positive Expression of human leukocyte antigen (HLA)-A2 Patients must have one related donor who is human leukocyte antigen (HLA) mismatched in the GVHD direction at two or more HLA loci Has a suitable human leukocyte antigen (HLA) haploidentical donor available Human leukocyte antigen (HLA)-A2 positive by deoxyribonucleic acid (DNA) sequence analysis (by history or as part of this study); HLA testing can be done at local labs Donor is blood-related and human leukocyte antigen (HLA)-haploidentical to the recipient 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*0201 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*02+ by serology by an ASHI accredited laboratory; Leukocyte count <3000/µL, Has a potentially suitable human leukocyte antigen (HLA) haploidentical donor available Patients must be human leukocyte antigen (HLA) A2 positive by polymerase chain reaction (PCR) typing Human leukocyte antigen (HLA) type A0201 or A2402 Patients must be human leukocyte antigen (HLA)-A*02:01-positive Human leukocyte antigen (HLA)-A2 positive. Any human leukocyte antigen (HLA) type (historic HLA typing is permitted) Patients must be human leukocyte antigen (HLA)-A*0201 positive Human leukocyte antigen (HLA)-A0201 or HLA-A2402 Availability of a willing and suitable human leukocyte antigen (HLA) identical related donor Human leukocyte antigen (HLA)-identical sibling donor Participants must be human leukocyte antigen (HLA)-A2 positive Presence of human leukocyte antigen (HLA) antibodies Availability of human leukocyte antigen (HLA)-identical sibling donor