Inclusion Criteria: 1. Subjects with hematologic malignancies for whom allogeneic stem cell transplantation is deemed clinically appropriate. Eligible diseases and stages include: - Acute lymphoblastic leukemia (including T lymphoblastic leukemia) in complete remission. - Remission is defined as < 5% blasts with no morphological characteristics of acute leukemia in a bone marrow with > 20% cellularity. - Acute myelogenous leukemia in high risk first CR or second or subsequent CR. - High risk first CR is defined by but is not limited to at least one of the following factors: greater than 1 cycle of induction chemotherapy to achieve CR, prior myelodysplastic syndrome (MDS), presence of Flt3 abnormalities, FAB M6 or M7 subtypes of leukemia, or adverse cytogenetics. - Remission is defined as < 5% blasts with no morphological characteristics of acute leukemia (e.g. Auer Rods) in a bone marrow with > 20% cellularity. - Biphenotypic/Undifferentiated leukemia in first or subsequent CR (same definition of CR as for ALL/AML). - Non-Hodgkin's lymphoma (T-cell, large cell or mantle cell) or Hodgkin's lymphoma in second or subsequent complete remission (CR) or in partial remission (PR) with documented chemosensitivity. In addition, marginal zone lymphoma or follicular lymphoma that has progressed after ≥ 2 therapies (excluding single-agent rituximab). No history of prior myeloablative procedure. 2. Lack of suitable 5-6/6 HLA-matched related or (if institutional guidelines dictate) suitable 8/8 HLA-A, B, C, DRß1 matched unrelated donor; or unrelated donor not available within appropriate timeframe. 3. Age 15-55 years. 4. Eastern Cooperative Oncology Group (ECOG) performance status 0-2. 5. Signed IRB approved Informed Consent Form (ICF). Exclusion Criteria: 1. History of prior allogeneic transplantation 2. Cardiac disease: symptomatic congestive heart failure or evidence of left ventricular dysfunction (Ejection fraction < 40%) as measured by gated radionuclide ventriculogram or echocardiogram; active angina pectoris, or uncontrolled hypertension; history of myocardial infarction with depressed ejection fraction. 3. Pulmonary disease: symptomatic chronic obstructive lung disease, symptomatic restrictive lung disease, or corrected DLCO of < 50% of predicted, corrected for hemoglobin. 4. Renal disease: serum creatinine > 2.0 mg/dl and calculated creatinine clearance < 40 mL/min. 5. Hepatic disease: serum bilirubin > 2.0 mg/dl (except in the case of Gilbert's syndrome or ongoing hemolytic anemia), SGOT or SGPT > 5 x upper limit of normal. 6. Neurologic disease: symptomatic leukoencephalopathy, active CNS malignancy or other neuropsychiatric abnormalities believed to preclude transplantation. 7. HIV antibody. 8. Uncontrolled infection. 9. Pregnancy or breast feeding mother. 10. Inability to comply with the requirements for care after allogeneic stem cell transplantation.