This study is a randomized, open-label, multicenter, prospective, comparative trial of granulocyte colony stimulating factor (G-CSF) mobilized peripheral blood stem cells verses marrow from unrelated donors for transplantation in patients with hematologic malignancies. The study will be comparing two-year survival probabilities between process has many steps after a matched donor is found. The donor also needs to give permission to join this study. Participants will be randomized to either peripheral blood stem cells (PBSC) or bone marrow. The kind of transplant the participant receives is random. Neither the participant nor the doctor may choose the type of transplant.