Many studies of allogeneic marrow transplantation have shown that a higher dose of marrow cells correlates with more robust hematopoietic engraftment and lower mortality from infectious complications. Peripheral blood stem cells (PBSC) collected after mobilization with granulocyte colony stimulating factor (G-CSF) contain a larger number of CD34-positive (CD34) progenitors and total cells than bone marrow. These observations led to the hypothesis that transplantation of PBSC would lead to lower mortality compared to transplantation of marrow. In addition, PBSC grafts have a higher T cell content, predicting a possibly more powerful anti-leukemia effect. However, the higher T cell content of PBSC may also lead to increased incidence and severity of acute and chronic graft-versus-host disease (GVHD). This concern is especially serious when the donor is unrelated to the recipient. This prospective, randomized, multicenter clinical trial of unrelated donor transplantation will test the hypothesis that transplantation of PBSC leads to similar patient survival compared to transplantation of marrow.