The primary purpose is to determine the ability of CD34+ selection and T cell depletion using the CliniMACS® device to prevent severe acute graft-versus-host disease (GVHD) in patients receiving a stem cell transplant from an alternative (unrelated and mismatched related) donor. The secondary objectives include evaluation of engraftment, immune recovery, and post-transplant infections. Patients requiring stem cell transplants for either malignant (cancerous) or non-malignant disease will be included in the study. The recipients will be grouped into one of two groups based on whether the donor is mismatched related (Cohort A) or unrelated (Cohort B). The patient will receive a conditioning regimen including chemotherapy drugs and/or total body irradiation based on the disease for which the transplant is performed.