 Research participants will be carefully monitored for engraftment, chimerism, incidence and severity of acute and chronic GvHD, regimen-related toxicity, characteristics of hematopoietic and immune reconstitution and ultimate survival and disease-free survival. This phase II trial is designed to investigate the feasibility and safety of a chemotherapy-based cytoreductive regimen plus a T-cell depleted peripheral blood stem cell PBSC or bone marrow stem cell transplant BMT for the treatment of high risk patients with advanced stages of hematologic malignancies. The majority of research participants will receive grafts derived from PBSC and will be the focus of the trial. The study population will be segmented into three research participant groups based on the type of donors used for the hematopoietic stem cell graft: 1 HLA-identical sibling or related donor, 2 HLA-compatible unrelated donor, and 3 HLA-mismatched related donor. 