The following criteria is provided for health care professionals.

Inclusion Criteria:

Age, Graft Cell Dose and Graft HLA Criteria

• Subjects must be <70 years old. Subjects ages ≥ 70 and ≤ 75 may be eligible if they have a Co-Morbidity Scoring (HCT-CI) score ≤ 2.
• The UCB graft is matched at 4-6 HLA-A, B, DRB1 antigens with the recipient.
• Patients co-enrolled in MT-2006-01 Phase I Study of Infusion of Umbilical Cord Blood Derived CD25+CD4+ T-Regulatory (Treg) Cells after Non-Myeloablative Cord\Blood Transplantation will receive grafts composed of 2 UCB units.

Disease Criteria:

• Acute Leukemias:
  • Acute myeloid leukemia: high risk complete remission 1 (CR1) (as evidenced by preceding myelodysplastic syndrome (MDS), high risk cytogenetics such as those associated with MDS or complex karyotype, > 2 cycles to obtain CR or erythroblastic and megakaryocytic); second or greater CR.
  • Acute lymphoblastic leukemia/lymphoma: high risk CR1 as evidenced by high risk cytogenetics (e.g. t(9;22), t(1;19),t(4;11), other myeloid/lymphoid or mixed lineage leukemia [MLL] rearrangements, hypodiploidy or Ikaros family zinc finger 1 [IKZF1]), > 1 cycle to obtain CR or evidence of minimal residual disease (MRD). Patients in second or greater CR are also eligible.
• Burkitt's lymphoma in CR2 or subsequent CR
• Natural Killer cell malignancies
• Chronic myelogenous leukemia: all types except refractory blast crisis. Chronic phase patients must have failed or been intolerant to Gleevec
• Myelodysplastic syndrome:
• Large-cell lymphoma, Hodgkin lymphoma and multiple myeloma with chemotherapy sensitive disease that has failed or patients who are ineligible for an autologous transplant.
• Chronic lymphocytic leukemia/small lymphocytic lymphoma (CLL/SLL), marginal zone B-cell lymphoma, follicular lymphoma, which have progressed within 12 months of achieving a partial or complete remission. Patients who had remissions lasting > 12 months, are eligible after at least two prior therapies. Patients with bulky disease should be considered for debulking chemotherapy before transplant. Patients with refractory disease are eligible, unless has bulky disease and an estimated tumor doubling time of less than one month.
• Lymphoplasmacytic lymphoma, mantle-cell lymphoma, prolymphocytic leukemia are eligible after initial therapy if chemotherapy sensitive.
• Refractory leukemia or MDS.
• Bone marrow failure syndromes, except for Fanconi Anemia
• Myeloproliferative syndromes Patients who have undergone an autologous transplant >12 months prior to allogeneic transplantation

Adequate Organ Function and Performance Status

Exclusion Criteria:

• < 70 years with an available 5-6/6 HLA-A, B, DRB1 matched sibling donor
• Pregnancy or breastfeeding
• Evidence of human immunodeficiency virus (HIV) infection or known HIV positive serology
• Current active serious infection
• Unless in post-chemotherapy and radioimmunoconjugated antibody induced aplasia, when he/she would be eligible for Arm 3, patients with acute leukemia in morphologic relapse/ persistent disease defined as > 5% blasts in normocellular bone marrow OR any % blasts if blasts have unique morphologic markers (e.g. Auer rods) or associated cytogenetic markers that allows morphologic relapse to be distinguished are not eligible.
• Chronic myelogenous leukemia (CML) in refractory blast crisis
• Large cell lymphoma, mantle cell lymphoma and Hodgkin disease that is progressive on salvage therapy. Stable disease is acceptable to move forward provided it is non-bulky.
• Active central nervous system malignancy