If transplantation using mismatched unrelated donors or non-first-degree relatives could be performed with an acceptable toxicity profile, an important unmet need would be served. Towards this goal, the current study extends our platform of nonmyeloablative, partially HLA-mismatched BMT to the use of such donors, investigating up to several postgrafting immunosuppression regimens that incorporate high-dose Cy. Of central interest is the incorporation of sirolimus into this postgrafting immunosuppression regimen. The primary goal is to identify a transplant regimen associated with acceptable rates of severe acute GVHD and NRM by Day 100.