Umbilical cord blood is used as a source of hematopoietic stem cells for bone marrow reconstitution in patients who would be potential candidates for a bone marrow transplant from an unrelated marrow donor. The outcome of transplantation is obtained to assess cord blood myeloid and platelet engraftment, transplant related mortality, overall survival, graft vs. host disease and, for patients with leukemia, lymphoma or myelodysplasia, relapse.