AlloHCT From Matched Unrelated Donors in Pts w/ Advanced Hematologic Malignancies & Disorders

OBJECTIVES: Primary * To determine the survival at day 100 of patients with relapsed, refractory, or poor-risk hematological malignancies treated with four different preparative regimens followed by allogeneic hematopoietic stem cell transplantation (HSCT) using two unrelated umbilical cord blood (UCB) units. Secondary * To determine the incidence and timing of neutrophil engraftment in patients treated with these regimens. * To determine the incidence and timing of platelet engraftment in patients treated with these regimens. * To determine the incidence and severity of acute and chronic graft-versus-host-disease (GVHD) in patients treated with these regimens. * To determine the survival at day 180 in patients treated with these regimens. * To determine the disease-free survival in patients treated with these regimens. * To determine the incidence of primary and secondary engraftment failure in patients treated with these regimens. * To determine the incidence of transplantation-related complications (e.g., infection, veno-occlusive disease of the liver, or organ toxicity) in these patients. * To determine the incidence of post-transplantation-related lymphoproliferative disease, secondary myelodysplastic syndromes, or other secondary malignancies in these patients. * To determine the incidence of relapse in patients treated with these regimens. * To determine post-transplantation chimerism in patients treated with these regimens. * To determine immune reconstitution in patients treated with these regimens. OUTLINE: This is a multicenter study. * Preparative regimens: Patients are assigned to 1 of 4 preparative regimens. * Regimen 1 (for patients \< 50 years of age and no contraindication to fractionated total-body irradiation (FTBI): Patients undergo FTBI 2-3 times a day on days -9 to -6 for a total of 11 fractions. Patients also receive cyclophosphamide IV over 2 hours on days -5 and -4 and fludarabine phosphate IV on days -5 to -2. * Regimen 2 (for patients \< 50 years of age and unable to tolerate FTBI due to prior dose-limiting radiotherapy or significant cardiotoxicity): Patients receive a test dose of busulfan on day -10 and then dose adjusted busulfan IV 3-4 times daily on days -9 to -6, melphalan IV on days -5 and -4, and fludarabine phosphate IV on days -5 to -2. * Regimen 3\* (for patients unable to tolerate regimen 1 or 2; no age exclusion): Patients receive fludarabine phosphate IV on days -8 to -4 and cyclophosphamide IV over 2 hours on day -3 and undergo TBI (single dose) on day -2. * Regimen 4\* (for patients unable to tolerate regimen 1 or 2): Patients receive fludarabine phosphate IV on days -7 to -3 and melphalan IV on day -2. NOTE: \*Treating physician decides the choice between regimen 3 and 4 * Umbilical cord blood (UCB) transplantation: Patients receive 2 combined units of UCB IV on day 0. Patients also receive G-CSF IV or subcutaneously beginning on day 5 (or later) and continuing until blood counts recover. * Graft-versus-host-disease prophylaxis: Patients receive cyclosporine IV twice daily beginning on day -1 followed by a taper according to institutional guidelines. Patients also receive mycophenolate mofetil orally or IV beginning on day 0 and continuing until day 27 (or as clinically indicated). After completion of study therapy, patients are followed periodically.