In this trial, the investigators will utilize a rapid generation protocol for broad spectrum multivirus-specific T cells for infusion to recipients of allogeneic hematopoietic stem cell transplant (HSCT), who are at risk of developing EBV, CMV, adenovirus, HHV6, BKV and/or HPIV3, or with PCR/culture confirmed infection(s). These cells will be derived from HSCT donors, and the study agent will be assessed at each dose for evidence of dose-limiting toxicities (DLT).