In this research study, the investigators want to learn more about the use of donor-derived viral specific T-cells (VSTs) to treat viral infections that occur after allogeneic stem cell transplant. A viral specific T cell is a T lymphocyte (a type of white blood cell) that kills cells that are infected (particularly with viruses). Allogeneic means the stem cells come from another person. These VSTs are cells specially designed to fight the virus infections that can happen after a bone marrow transplant. The investigators are asking people who have undergone or will undergo an allogeneic stem cell transplant to enroll in this research study, because viral infections are a common problem after allogeneic stem cell transplant and can cause significant complications including death. Stem cell transplant reduces a person's ability to fight infections. There is an increased risk of getting new viral infections or reactivation of viral infections that the patient has had in the past, such as cytomegalovirus (CMV), Epstein-Barr virus (EBV), adenovirus (ADV), BK virus (BKV), and JC virus. There are anti-viral medicines available to treat these infections, though not all patients will respond to the standard treatments. Moreover, treatment of viral infections is expensive and time consuming, with families often administering prolonged treatments with intravenous anti-viral medications, or patients requiring prolonged admissions to the hospital. The medicines can also have side effects like damage to the kidneys or reduction in the blood counts, so in this study the investigators are trying to find an easier way to treat these infections.