The proposed study is based on our observation of paradoxical tumor regression after rejection of the donor graft in conjunction with the results of our murine experiments. We hypothesize that clinically meaningful responses can be achieved in patients with advanced malignancies with a transplant strategy using nonmyeloablative conditioning and related mismatched donor stem cell transplant where the intention will be to initially achieve mixed chimerism which will be followed by recipient lymphocyte infusion (RLI) in an attempt to deliberately reject the donor graft. This will lead to the development of novel transplant strategies for achieving antitumor effects without the risk of graft versus host disease (GVHD). This proposed protocol is a Pilot Study that will evaluate the safety of this outpatient transplant strategy, i.e., establishment of initial mixed chimerism followed by RLI for donor graft rejection, in patients with advanced lymphomas, and multiple myeloma.