Allogeneic hematopoetic stem cell transplantation (SCT) is frequently complicated by life threatening viral reactivation. Conventional antiviral therapy is suboptimal for cytomegalovirus (CMV), adenovirus (AdV) and Epstein-Barr virus (EBV) and nonexistent for BK virus (BKV). An alternative approach to prevent viral reactivation is to infuse virus-specific cytotoxic T cells (CTL) prepared from the donor early after SCT. Such multivirus-specific CTL cells (MVST) have been successfully used in a number of centers to prevent or treat CMV, Ad and EBV. Activity of BKV-reactive cells has not been studied. Multi virus-specific T cells (MVST) are donor lymphocytes that are highly enriched for viral antigens and expanded in vitro before infusion into the transplant recipient. Viral reactivation is a particular problem inT cell depleted SCT. Median time to CMV reactivation is estimated as 28 days post T-depleted transplant, but infusion of MVST within the immediate post-SCT period has not been previously studied. This protocol will be the first of a planned series of cellular therapies to be layered on our existing T lymphocyte depleted transplant platform protocol 13-H-0144.