This research project will focus on whether it is safe and effective to rely on donor cells to prevent relapse of leukemia, lymphoma, or other blood cancer after bone marrow stem cell transplant (SCT). For many patients with these conditions, a bone marrow transplant can be the only chance for cure. The standard type of bone marrow transplant involves giving very high-doses of chemotherapy and radiation to kill all the cancer cells followed by an infusion of bone marrow stem cells from a relative who is a bone marrow match. After the transplant the patient takes anti-rejection drugs for many months to prevent the donor’s immune cells from causing a severe reaction called Graft-versus-Host Disease (GVHD), which can even be fatal. However, for some patients with certain types of high-risk cancer even this intense treatment is not effective and the cancer relapses.