Umbilical Cord Blood Transplantation Using a Myeloablative Preparative Regimen for Hematological Diseases

Inclusion Criteria: * Eligible Disease Status * Acute Myeloid Leukemia (AML): high risk CR1 (as evidenced by preceding MDS, high risk cytogenetics, ≥ 2 cycles to obtain 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 \< 25% blasts in marrow after having failed one or more cycles of chemotherapy. * Acute Lymphocytic Leukemia (ALL): high risk CR1 as defined by cytogenetics (such as t(9;22), t (1:19), t(4;11), 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 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 de-bulking 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. * Myeloproliferative Syndromes * Availability of suitable UCB unit(s) * 0 to 55 years * Voluntary written consent (adult or parental/guardian) Exclusion Criteria: * previous irradiation that precludes the safe administration of TBI - Radiation Oncology will evaluate all patients who have had previous radiation therapy * chemotherapy refractory large cell and high grade NHL (ie progressive disease after \> 2 salvage regimens) * if ≤ 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 * pregnant or breastfeeding * HIV positive