• Two-year survival from the time of randomization.
  

• Two-year survival
  
• Neutrophil Engraftment > 500/mcl
  
• Primary graft failure
  
• Platelet engraftment > 20,000 and 50,000/mcl transfusion independent
  
• Acute GVHD of grades II-IV and III-IV
  

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. Participants will receive vaccinations for diphtheria, tetanus, hepatitis B and pneumococcus. Blood samples will be drawn as part of a participant's normal care. Blood samples will also be collected to see if infection fighting cells are working. These samples will be collected at 12 weeks, one year and then yearly until three years post transplant. Information will be collected from the participant's medical history. Participants will also have phone interviews conducted and questions will be asked about their physical and emotional health.

Participants taking part in this study may be followed for up to 5 years.