The PUMA Trial is a Trial of a Single ProHema Modulated-Cord Blood (CB) Unit as Part of a Double CB Transplant in Patients With Hematologic Malignancies.

Inclusion: 1. Patients with hematologic malignancies for whom allogeneic stem cell transplantation is deemed clinically appropriate. Eligible diseases and stages include: 1. Acute lymphoblastic leukemia (ALL) (including T lymphoblastic lymphoma) in complete remission (CR). • Remission is defined as \< 5% blasts with no morphological characteristics of acute leukemia in a bone marrow with \> 5% cellularity. 2. Myelodysplastic disease, International Prognostic Scoring System (IPSS) Intermediate-2 or High risk (e.g., refractory anemia with excess blasts (RAEB), refractory anemia with excess blasts in transformation (RAEBt) or refractory anemia with severe pancytopenia or high risk cytogenetics. Patients have to have received leukemia type induction chemotherapy within ≤ 3 months and with ≤ 10% blasts by a bone marrow aspirate; a single hypomethylating agent is not considered adequate cytotoxic chemotherapy; all subtypes except chronic myelomonocytic leukemia (CMML). 3. Acute myelogenous leukemia (AML) 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 one cycle of induction chemotherapy to achieve CR, prior myelodysplastic syndrome (MDS), presence of fms-like tyrosine kinase 3 (FLT3) abnormalities, French-American-British (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 \> 5% cellularity. * AML arising from myelofibrosis is not permitted. 4. Biphenotypic/undifferentiated leukemia in first or subsequent CR (same definition of CR as for ALL/AML). 5. Chronic myelogenous leukemia (CML) with prior exposure to cytotoxic chemotherapy for the treatment of blast phase or with demonstrated intolerance to at least 2 tyrosine kinase inhibitors. 6. Non Hodgkin's lymphoma (T cell, large cell or mantle cell) or Hodgkin's lymphoma in second or subsequent 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). * If the myeloablative conditioning regimen is selected, a history of prior myeloablative procedure is not allowed. * If the reduced intensity conditioning regimen is selected, and the subject has had a prior autologous transplant, it must have taken place \> 3 months from anticipated Day 0 visit. 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. Both cord blood units (CBUs) are qualified by Fate Therapeutics 4. Age 15 to 55 years (myeloablative regimen) or 15 to 65 years (reduced intensity regimen) 5. Body weight \> 45 kg 6. Investigator selection of conditioning regimen (myeloablative or reduced intensity) 7. Eastern Cooperative Oncology Group (ECOG) Performance Status 0 to 2. 8. Signed Institutional Review Board (IRB) approved Informed Consent Form (ICF). Exclusion: 1. History of prior allogeneic transplantation 2. Cardiac disease: symptomatic congestive heart failure or evidence of left ventricular dysfunction (Ejection fraction \< 50%) 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 diffusing capacity for carbon monoxide (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), aspartate aminotransferase (SGOT) or alanine aminotransferase (SGPT) \> 5 × upper limit of normal. 6. Neurologic disease: symptomatic leukoencephalopathy, active central nervous system (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.