The primary purpose of this phase I/II study is to evaluate whether partially matched, ≥2/6 HLA-matched, viral specific T cells have efficacy against adenovirus, CMV, and EBV, in subjects who have previously received any type of allogeneic HCT or solid organ transplant (SOT), or have compromised immunity. Reconstitution of anti-viral immunity by donor-derived cytotoxic T lymphocytes has shown promise in preventing and treating infections with adenovirus, CMV, and EBV. However, the weeks taken to prepare patient-specific products, and cost associated with products that may not be used limits their value. In this trial, we will evaluate viral specific T cells generated by gamma capture technology. Eligible patients will include HCT and/or SOT recipients, and/or patients with compromised immunity who have adenovirus, CMV, or EBV infection or refractory viremia that is persistent despite standard therapy. Infusion of the cellular product will be assessed for safety and efficacy.