This is a Phase III randomized, open label, multicenter clinical trial sponsored by the National Marrow Donor Program (NMDP) and the National Institutes of Health (NIH). The objective of the trial is to test the null hypothesis that there is no difference in overall survival after PBSC versus marrow transplants from HLA compatible unrelated donors. The study will compare G-CSF-mobilized PBSC transplantation with bone marrow transplantation from HLA-compatible unrelated donors for patients with leukemia, myelodysplastic or myeloproliferative syndromes. Conditioning and GVHD prophylaxis regimens will vary by center and within centers, however, the center must declare before randomization what regimens will be used for each patient. The primary endpoint of this trial is 2-year survival following randomization. Secondary analyses will consider neutrophil and platelet recovery, acute and chronic GVHD, time off all immunosuppressive therapy, relapse, infections, adverse events and immune reconstitution. The trial will include evaluation of patient and donor quality of life, composition of the graft, and immune reconstitution. Accrual is anticipated for 3 years with a follow-up period of 3 years.