Inclusion Criteria: - Age equal or less than 65 years old and equal to or greater than 18 years old. - Patients with the diagnosis of MDS or AML with fewer than 5% myeloblasts in the bone marrow and no leukemic myeloblasts in the peripheral blood on morphologic analysis performed within 30 days of start of the conditioning regimen enrollment. - For patients receiving treatment of their MDS or AML prior to transplantation: a)Interval between the start of the most recent cycle of conventional cytotoxic chemotherapy and enrollment must be at least 30 days; b)Interval between completing treatment with a hypomethylating agent or other non-cytotoxic chemotherapy and enrollment must be at least 10 days. - Patients must have a related or unrelated bone marrow or peripheral blood donor who is HLA-matched at 7 or 8 of 8 HLA-A, -B, -C and -DRB1 at high resolution using DNA-based typing. - HCT-Specific Comorbidity Index Score (HCT-CI) less than or equal to 4. - Organ function: a) Cardiac function: Ejection fraction greater than or equal to 40%; b) Hepatic function: total bilirubin less than or equal to 2 times the upper limit of normal and ALT and AST less than or equal to 3 times the upper limit of normal.; c)Pulmonary function: DLCO greater than or equal to 40% and FEV1 greater than or equal to 50% (corrected for hemoglobin. - Creatinine clearance greater than or equal to 50mL/min based on the Cockcroft-Gault formula. - Signed informed consent. Exclusion Criteria: - Prior allograft or prior autograft. - Symptomatic coronary artery disease. - Leukemia involvement in the CNS within 4 weeks of enrollment for patients with a history of prior CNS leukemia involvement (i.e., leukemic blasts previously detected in the cerebral spinal fluid). - Karnofsky Performance Score less than 70. - Patients receiving supplemental oxygen. - Planned use of DLI therapy. - Patients with uncontrolled bacterial, viral or fungal infections (undergoing appropriate treatment and with progression of clinical symptoms). - Patients seropositive for the human immunodeficiency virus (HIV). - Patients with prior malignancies, except resected basal cell carcinoma or treated cervical carcinoma in situ. Cancer treated with curative intent greater than 5 years previously. Cancer treated with curative intent less than 5 years previously will not be allowed unless approved by the Protocol Officer or one of the Protocol Chairs. - Females who are pregnant or breastfeeding. - Fertile men and women unwilling to use contraceptive techniques during and for 12 months following treatment.