Multi-institutional Prospective Research of Expanded Multi-antigen Specifically Oriented Lymphocytes for the Treatment of VEry High Risk Hematopoietic Malignancies

Recipient Inclusion Criteria to Enter Protocol: * Aged 6 months to 80 years. * Anticipated myeloablative or non-myeloablative allogeneic hematopoietic stem cell transplant. * Patients with high risk AML and MDS who have received or will receive an allo-HSCT and have not had hematologic relapse of disease. * Karnofsky/Lansky score of ≥ 50. * Agree to use contraceptive measures during study protocol participation (when age appropriate). * Patient or parent/guardian capable of providing informed consent. * T cell chimerism \> 94% if collected from recipient of allo-HSCT Recipient Exclusion Criteria to Enter Protocol: * Patients with uncontrolled infections. * Current evidence of GVHD \> grade 2 or bronchiolitis obliterans syndrome, sclerotic GVHD, or serositis. * Pregnancy (female of childbearing potential). Recipient Inclusion Criteria for TAA-T Administration: * Patients with high risk AML and MDS who have received an allo-HSCT and have not had hematologic relapse of disease. * Steroids less than 0.5 mg/kg/day prednisone or equivalent in the context of no escalation of treatment within the preceding 2 weeks * Karnofsky/Lansky score of ≥ 50. * Bilirubin \< 2.5 mg/dL, AST/ALT \<5x upper limit of normal, Serum creatinine \< 1.0 or 2x the upper limit of normal (whichever is higher). * Pulse oximetry of \> 90% on room air. * Absolute neutrophil count \> 250/ µL (may be supported with Granulocyte colony-stimulating factor (GCSF)). * Agree to use contraceptive measures during study protocol participation (when age appropriate). * Patient or parent/guardian capable of providing informed consent. * LVEF \> 50% or LVSF \> 27% (performed within the last 6 months) if history of TBI \>500 cGy for arm A and B. * Total chimerism \> 50%; or if cancer cells preclude this, donor T cell chimerism \> 50% (performed within the last 6 months). Recipient Exclusion Criteria for TAA-T Administration: * Patients who received ATG, Campath, or other T cell immunosuppressive monoclonal antibodies within 28 days prior to TAA-T infusion. * No investigational therapies (under IND, not extensively studied in the current clinical context) within 28 days prior to TAA-T infusion. * Uncontrolled infections. * Active Bronchiolitis obliterans syndrome, sclerotic GVHD, or serositis. * Active acute GVHD or chronic GVHD requiring escalation of treatment within preceding 2 weeks of any grade is exclusion for Arm C patients. * Pregnancy or lactating (female of childbearing potential). * Patients who have or will be receiving 2nd allogeneic HSCT Donor Inclusion Criteria: * Donors for allogeneic (i.e. HLA matched or mismatched related or unrelated) stem cell transplants who have undergone eligibility evaluation as per FDA regulations outlined in 21 CFR 1271 subpart C. If a donor has been chosen for the transplant based on urgent medical need, that same donor will also be used for TAA-T generation provided that there are no new reasons for ineligibility since the transplant donor evaluation. * Aged 6 months to 80 years. * Donor or guardian of pediatric capable of providing informed consent. * Donor must have completed infectious Disease (ID) testing up to 7 days before or after the collection of blood from the donor (related or unrelated) for TAA-T manufacturing. The following tests will be performed: * HBsAg * HB Core antibody * HIV1/2 NAT * Syphilis (T. Pallidum IgG) * HTLV I/II * CMV total * HBV/HCV NAT * West Nile Virus NAT. * Cruz (Chagas) antibody * Hepatitis C * Female donors of childbearing age must have a negative pregnancy test within 7 days of blood collection for TAA-T manufacturing. Donor Exclusion Criteria: * Donation of cells would pose a physical or psychological risk to the donor. * Female donors of childbearing age who are known to be pregnant.