Myeloablative Umbilical Cord Blood Transplantation in Hematological Diseases

Inclusion Criteria: * Acute myeloid leukemia (AML): high risk CR1 (as evidenced by preceding myelodysplastic syndrome \[MDS\], high risk cytogenetics, ≥ 2 cycles to obtain complete remission \[CR\], erythroblastic or megakaryocytic leukemia; CR2+. All patients must be in CR as defined by hematological recovery, AND \<5% blasts by light microscopy within the bone marrow with a cellularity of ≥15%. * Very high risk pediatric patients with AML. Patients \<21 years, however, are eligible with (M2 marrow) with \< or = 25% blasts in marrow after having failed one or more cycles of chemotherapy. This group of patients will be analyzed separately. * Acute lymphocytic leukemia (ALL): high risk CR1 \[t(9;22), t (1:19), t(4;11) or other MLL rearrangements\] hypodiploidy, or IKZF1 abnormalities), DNA index \< 0.81, \> 1 cycle to obtain CR or presence minimal residual disease (MRD). Patients in CR2+ are eligible. All patients must be in CR as defined by hematological recovery, AND \<5% blasts by light microscopy within the bone marrow with a cellularity of ≥15%. * Very high risk pediatric patients with ALL. patients \<21 years are also considered high risk CR1 if they had M2 or M3 marrow at day 42 from the initiation of induction or M3 marrow at the end of induction. They are eligible once they achieved a complete remission * Chronic myelogenous leukemia (CML) excluding refractory blast crisis. To be eligible in first chronic phase (CP1) patient must have failed or be intolerant to imatinib mesylate. * Plasma Cell leukemia after initial therapy, who achieved at least a partial remission * Advanced myelofibrosis * Myelodysplasia (MDS) IPSS Int-2 or High risk (i.e. RAEB, RAEBt) or refractory anemia with severe pancytopenia or high risk cytogenetics. Blasts must be \< 10% by a representative bone marrow aspirate morphology. * Chronic lymphocytic leukemia/small lymphocytic lymphoma (CLL/SLL), marginal zone B-cell lymphoma or follicular lymphoma are eligible if there was disease progression/relapse within 12 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 (nodal mass greater than 5 cm) should be considered for debulking chemotherapy before transplant. * Lymphoplasmacytic lymphoma, mantle-cell lymphoma, prolymphocytic leukemia are eligible after initial therapy in CR1+ or PR1+. * Large cell NHL \> CR2/\> PR2. Patients in CR2/PR2 with initial short remission (\<6 months) are eligible. * Lymphoblastic lymphoma, Burkitt's lymphoma, and other high-grade NHL after initial therapy if stage III/IV in CR1/PR1 or after progression if stage I/II \< 1 year. * Multiple myeloma beyond PR2. Patients with chromosome 13 abnormalities, first response lasting less than 6 months, or β-2 microglobulin \> 3 mg/L, may be considered for this protocol after initial therapy. * Recipients must have a Karnofsky score (adults) ≥ 80 % or Lansky score ≥ 50% (pediatrics), and proper organ function. Exclusion Criteria * Active infection at time of transplantation * History of human immunodeficiency virus (HIV) infection * Pregnant or breast feeding. * Chemotherapy refractory large cell and high grade NHL * If \< or = 18 years old, prior myeloablative transplant within the last 6 months. If \>18 years old prior myeloablative allotransplant or autologous transplant * Extensive prior therapy including \> 12 months alkylator therapy or \> 6 months alkylator therapy with extensive radiation. * Patients who have received Y-90 ibritumomab (Zevalin) or I-131 tostumomab (Bexxar), as part of their salvage therapy.