The purpose of this research is to study how immune cells (called T cells) that cause graft-versus-host disease (GVHD) can best be selectively separated from other T cells and removed from the cells that will be returned to the cancer patient's body. These other T cells may protect against infection when given to patients after a stem cell transplant. The removal of cells that cause GVHD would allow doctors to safely give back the T cells that protect against infection, without the risk of GVHD.