Patients must be positive for at least one of the following human leukocyte antigens (HLA): a. HLA-A*:, b. HLA-A*:, c. HLA-A*: DONOR: Related or unrelated volunteer donors who are mismatched with the recipient within one of the following limitations:\r\n* Mismatch for one HLA class I antigen with or without an additional mismatch for one HLA-class I allele, but matched for HLA-DRB and HLA-DQ, OR\r\n* Mismatched for two HLA class I alleles, but matched for HLA-DRB and HLA-DQ, OR\r\n* HLA class I HLA-A, -B, -C allele matched donors allowing for any one or two DRB and/or DQB antigen allele mismatch Subject is HLA-A*:, HLA-A*:, and/or HLA-A*: positive as determined by a central laboratory. (This determination will be made under a pre enrollment screening informed consent form [ICF]. There are no restrictions on the timing of HLA typing for screening and data can be taken from subjects' records). HLA-mismatch at the HLA-A, -B, -C, and - DRB loci. Note, HLA-DQ mismatches are permissible The patient must test positive for HLA-A (tested by a CLIA approved laboratory; only the\r\n HLA A*: subtype is eligible) Subject is HLA-A*: and/or HLA-A*: positive. Subject is HLA-A*: in either allele, HLA-B*: and/or HLA-B*: positive. Subject has any A* null allele (designated with an \N\, e.g. A*:N) as the sole HLA-A* allele. Patients must be positive for HLA-A based on flowcytometry or genotyping A minimum genotypic identical match of / is required, as determined by high resolution typing, at least one allele of each of the following genetic loci: HLA-A, HLA-B, HLA-Cw, and HLA- DRB. HLA-A* (HLA-A.) positivity by molecular subtyping Patients must express HLA-A* Patients must have an adult donor for HCT who is adequately HLA matched by institutional standards (includes HLA-matched related or unrelated donors, and HLA-mismatched family donors, including haploidentical donors) and is either: \r\n* HLA-A* positive and HA-(H) negative (RS_: G/G) or \r\n* HLA-A* negative Patients must have an adult donor for HCT who is adequately HLA matched by institutional standards (includes HLA-matched related or unrelated donors, and HLA-mismatched family donors, including haploidentical donors) and is either: \r\n** HLA-A* positive and HA-(H) negative (RS_: G/G) or\r\n** HLA-A* negative HLA-MISMATCHED UNRELATED DONOR: Unrelated volunteer donors who are mismatched with the recipient within one of the following limitations:\r\n* Mismatch for one HLA class I antigen with or without an additional mismatch for one HLA-class I allele, but matched for HLA-DRB and HLA-DQ, OR \r\n* Mismatched for two HLA class I alleles, but matched for HLA-DRB and HLA-DQ\r\n* HLA class I HLA-A, -B, -C allele matched donors allowing for any one or two DRB and/or DQB antigen/allele mismatch HLA-A* (HLA-A.) positivity by molecular subtyping Patients must have the HLA-A*: allele Patients must be HLA-A* positive Subject is HLA-A*: positive in either allele; Subject has HLA-A*: as the sole HLA-A* allele (e.g., a subject with HLA alleles A*: and A*: is eligible); or Subject has any A* null allele (designated with an \N\, e.g., A*:N) as the sole HLA-A* allele Subject is HLA-A*: or HLA-A*: positive. Subject is HLA-A*:, HLA-B*: and/or HLA-B*: positive. Must be HLA- A*: positive; (retesting is not required for patients who have previous documented HLA- A*: positivity) HLA phenotype positive. Patients must be HLA-A* positive HLA-A*: expression Lack of HLA-A*: expression Patient Human Leukocyte Antigen (HLA) typing should demonstrate HLA-A*, and/or HLA-A*, and/or HLA-A* restriction. Patients with HLA-A alleles not belonging to any of the following subtypes: HLA-A*, or HLA-A*, or HLA-A*. Subjects must express HLA-A phenotype as assessed serologically. HLA-A negative patients All study participants must have one of the HLA alleles: HLA-A*, HLA-A*, HLAA*, or HLA-A* Patients must be HLA-A* positive Patients must express HLA-A* HLA-A* positive by Central Assay Be HLA-A*+ as determined by Central Laboratory; Patient Human Leucocyte Antigen (HLA) typing should demonstrate HLA-A*, and/or HLA-A*, and/or HLA-A* restriction. Patients without HLA-A, or HLA-A*, or HLA-A* restriction. Subjects must be HLA-A positive HLA phenotype positive. Note: Patients who were previously HLA-typed for participation in other Immatics' sponsored clinical trials and were HLA phenotype positive may enter IMA- main screening Expression of HLA-A*. Expression of HLA-A*. Be HLA-A positive Patient is HLA-A+ HLA-A* positive as determined by deoxyribonucleic acid (DNA) allele-specific polymerase chain reaction (PCR) assay; HLA restriction is not required for cohorts and HLA-A* positive as determined by deoxyribonucleic acid (DNA) allele-specific polymerase chain reaction (PCR) assay\r\n* For cohort after amendment and cohort , there is no HLA restriction The donor and recipient must be identical, as determined by high resolution typing, on at least one allele of each of the following genetic loci: HLA-A, HLA-B, HLA-Cw, HLA- DRB and HLA-DQB. Subject must be HLA-A*:, HLA-A*: and/or HLA-A*: positive. Human leukocyte antigen (HLA)-A* (HLA-A.) positivity by molecular subtyping\r\n* Patients with HLA-A* (HLA-A.) positivity by molecular subtyping may be eligible if there is demonstration that they can correctly present the MART- - epitope as stimulators for interferon (IFN)-gamma production by MART- F TCR transgenic cells Human leukocyte antigen (HLA)-A*, HLA-A*, HLA-A* or HLA-A* positive patients Must be HLA A*, HLA-A*, and/or HLA-A* positive by high resolution testing. The patient must express HLA class I allele HLA-A* for NY-ESO-/LAGE. Blood HLA-A phenotype HLA-A*: positive Patient must be HLA-A*+ and/or HLA-A*+ Subject must be HLA A*:, HLA A*: and/or HLA-A*: positive. HLA A* high resolution, -digit type is required at HLA-A to ensure A* status HLA-A+ or HLA-A+ or HLA-A+ or HLA-A+ UNRELATED DONOR: an HLA / or / HLA A, B and C (class I) and HLA DRB (class II) alleles, or / or / HLA A, B and C (class I) and HLA DRB and DQB (class II) alleles will be required for study entry HLA-A*:, HLA-A*:, and/or HLA-A*: by high resolution testing at a local or central laboratory Patients must be HLA-A positive HLA-A negative patients HLA-A or HLA-A haplotype HLA-DPB*: positive Positive for HLA-A*:or HLA-A*: allele. Positive for any of the HLA-A* allele other than HLA-A*: or HLA-A*: or the following alleles: HLA-A*:, HLA-C*: or HLA-B*:.