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