Umbilical Cord Blood Transplantation From Unrelated Donors

Inclusion Criteria: * Appropriate diagnosis: Patients must have a disease or syndrome amenable to therapy with hematopoietic stem cell transplantation. Diagnoses include, but are not limited to: * Congenital and Other Non-malignant Disorders: * Immunodeficiency disorders (e.g. Severe Combined Immunodeficiency, Wiskott-Aldrich Syndrome) * Congenital hematopoietic stem cell defects (e.g. Chediak-Higashi Syndrome, Congenital Osteopetrosis, Osteogenesis Imperfecta) * Metabolic disorders (e.g. Hurler's Syndrome) * Severe aplastic anemia * High-Risk Leukemia: * Acute Myelogenous Leukemia * Refractory to standard induction therapy (more than 1 cycle required to achieve remission) * Recurrent (in CR ≥ 2) * Treatment-related AML or MDS * Evolved from myelodysplastic syndrome * Presence of FLT3 abnormalities * FAB M6 or M7 * Adverse cytogenetics * Myelodysplastic Syndrome * Acute Lymphoblastic Leukemia including T lymphoblastic leukemia: * Refractory to standard induction therapy (time to CR \>4 weeks) * Recurrent (in CR ≥ 2) * WBC count \>30,000/mcL at diagnosis * Age \>30 at diagnosis * Adverse cytogenetics, such as t(9:22), t(1:19), t(4:11), and other MLL rearrangements. * Chronic Myelogenous Leukemia in accelerated phase or blast crisis * Biphenotypic or undifferentiated leukemia * Burkitt's leukemia or lymphoma * Lymphoma: * Large cell, Mantle cell, Hodgkin lymphoma refractory or recurrent, chemo-sensitive, and ineligible for an autologous stem cell transplant or previously treated with autologous SCT * Marginal zone or follicular lymphoma that is progressive after at least two prior therapies * Multiple Myeloma, recurrent following high-dose therapy and autologous SCT or ineligible for an autologous HSCT * Solid tumors, with efficacy of allogeneic HSCT demonstrated for the specific disease and disease status * Adequate organ function: * Cardiac - LVEF \>45%, or shortening fraction \>25%, Absence of congestive heart failure or conduction disturbances with high risk for sudden death * Pulmonary - DLCO (corrected for hemoglobin), FEV1 and FVC ≥ 50% predicted; * Renal - serum Cr \< 1.5 times the upper limit of normal for age or GFR ≥ 50 ml/min/1.73m2 * Hepatic - total bilirubin level \< 2 times the upper limit of normal (except for patients with Gilbert's syndrome or hemolysis); if the primary disease process is causal, this criterion will be reconsidered. ALT, AST, and Alkaline phosphatase ≤ 5 times upper limit of normal. * Performance Status Karnofsky or Lansky score ≥ 70%. * Informed Consent must be obtained prior to initiating conditioning therapy. * Receipt of viable cord blood product(s), single or dual, must be confirmed with the stem cell processing laboratory prior to initiating conditioning therapy. Exclusion Criteria: * Availability of 10/10 or 9/10 HLA-matched related or unrelated donor within a reasonable timeframe dictated by the clinical urgency of the transplant * Autologous HSCT \< 6 months prior to proposed UCB transplant * Pregnant or breast feeding * Current uncontrolled infection * Evidence of HIV infection or positive HIV serology