 Participants will receive one of the two conditioning regimens described in the protocol, at the discretion of the transplant physician. The transplant physician must choose among these regimens prior to the participants assignment to the GVHD prophylaxis treatment. Conditioning regimens will vary by center, but will be the same for all participants at each center. Stem cell donors will donate peripheral blood stem cells according to local institutional practices. Peripheral blood stem cells will not be manipulated or T-depleted prior to administration. Standard post-transplant care will be administered. Participants will be randomly assigned to one of two GVHD prophylaxis regimens and will be followed for the endpoints of interest. 