The objective of this study is to make T-cell depleted stem cells from a family member who is a half match (haplo-identical) available on an expanded access basis to patients receiving one or two unrelated cord blood transplants who are at a higher risk of not engrafting in a safe amount of time. The purpose of the related stem cells is to give the bone marrow a "jump start" towards recovery. Ultimately, the cord blood cells will grow and permanently rescue the bone marrow.

The primary purpose of the study is to provide expanded access of T-cell depleted haplo-identical stem cells for patients receiving allogeneic transplantation from a related haplo-identical donor and an unrelated, umbilical cord blood (UUCB) unit(s) for the treatment of high risk malignancies and non-malignant disorders. The T-cell depleted haplo-identical stems cells are intended to facilitate early, short-term myeloid engraftment with the primary goal of minimizing early infections and other non-relapse mortality while the UUCB cells engraft as the durable and permanent graft. Patients with high risk or refractory malignancies, or non-malignant disorders amenable to stem cell transplantation therapy but lacking conventional related or unrelated donors will be eligible for this protocol.

Inclusion Criteria:

• Have a consenting related haplo-identical (3/6, 4/6, or 5/6 if DRB1 mismatch) stem cell donor.
• Have one or two available 4, 5, or 6/6 antigen matching unrelated UCB unit(s) that will deliver a total cell dose >3.0 x 10e7 cells/kg. Patients who do not have a single UCB unit that will deliver the minimum required cell dose, two partially HLA-matched UCB units which together meet the minimum cell dose requirement, can be used for 1 transplant. These units must be HLA-matched minimally at 4 of 6 HLA-A and B (at intermediate resolution by molecular typing) and DRB1 (at high resolution by molecular typing) loci with the patient, and HLA-matched at 3 of 6 HLA- A, B, DRB1 loci with each other (using same resolution of HLA typing as indicated above). There is no limitation on maximum cell dose.
• Have a high risk or refractory malignancy, or non-malignant disorder amenable to stem cell transplantation therapy.
• Meet eligibility requirements for allogeneic transplant per institutional standard practices.
• Have given written informed consent according to FDA guidelines (or consent of parent/legal guardian as applicable).
• Be <65 years of age at the time of study enrollment.

Exclusion Criteria:

• Have a consenting 8/8 or 10/10 allele matched, consenting, related or unrelated hematopoietic stem cell transplant (HSCT) donor.
• Have a life expectancy of less than 3 months.
• Have uncontrolled infections at time of cytoreduction.