The primary objective is to determine the safety and feasibility of administering R-MVST cells to patients with refractory viral reactivation and/or symptomatic disease caused by Epstein Barr Virus (EBV), cytomegalovirus (CMV), adenovirus (ADV) or BK virus. R-MVST cells will be generated on-demand from the closest partially human leukocyte antigen (HLA)-matched (minimum haploidentical) healthy donors or from the original allo-transplant donor if available. The investigator will closely monitor the recipients for potential toxicities including graft-versus-host disease (GVHD) post-infusion.

Secondary objectives are to determine the effect of R-MVST infusion on viral load, possible recovery of antiviral immunity post-infusion and for evidence of clinical responses and overall survival. Recipients will be monitored for secondary graft failure at day 28 post R-MVST infusion.

Starting from childhood, majority of healthy humans are exposed to common viruses such as CMV, EBV, BK and related human polyomaviruses and herpes viruses. Under normal circumstances those infections are well controlled by the adaptive immune system, but never eliminated. Instead, they are fairly inactive and produce relatively few consequences or symptoms. However, when T cell mediated immunity is suppressed, those dormant viruses reactivate and can cause a significant end-organ or severe systemic syndrome. This viral reactivation contributes to morbidity and mortality in recipients of allogeneic stem cell transplant (HCT) and solid organ transplants (SOT), and can affect many other patients who receive immunosuppressive therapies or have underlying pathology that affects T cell function, including patients with autoimmune diseases, congenital immunodeficiencies or HIV/AIDS. As a result of a weakened immune response, conventional antiviral prophylaxis or treatment with acyclovir and ganciclovir/foscarnet (for CMV) or rituximab (against EBV) are not always effective.

The main purpose of this study is to test whether giving an experimental cell product can treat the viral infection in patients who have conditions that cause poor function of their immune system, such as infections caused by viruses such as Cytomegalovirus (CMV), Epstein-Barr Virus (EBV), BK virus, or adenovirus. The cell product is called rapidly generated virus specific T cells or R-MVST.

Recipient Inclusion Criteria:

• Men and women ages 18 years or older of all ethnic groups will be eligible for the treatment
• Patients with history of HCT or SOT who demonstrate evidence of viral reactivation and/or infection manifesting as end-organ or systemic disease due to one or more of the following viruses: EBV, CMV, ADV or BK virus and suboptimal response to the standard of care therapy.
• Recurrent or Multiple Viral Infection. RVI defined as occurrence of more than one episode of reactivation that required intervention or symptomatic disease in recipient of allogeneic HCT that required standard of care treatment. MVI defined as more than one virus reactivating (defined by PCR positivity) or causing symptomatic systemic or end-organ disease. At least one of those viral reactivations required standard of care intervention. No standard of care therapy is defined for ADV and BK. Patients with multiple infections/reactivations will be eligible as long as at least one of those viral infections meet the criterium of "refractory".

Recipient Exclusion Criteria:

• Patients with other uncontrolled infections, except for CMV, EBV, ADV or BK. For bacterial infections, patients must be receiving definitive therapy and have no signs of progressing infection for 72 hours prior to the day of infusion. For fungal infections, patients must be receiving definitive systemic anti-fungal therapy and have no signs of progressing infection for 1 week prior to R-MVST infusion. Progressing infection is defined as hemodynamic instability attributable to sepsis or new symptoms, worsening physical signs or radiographic findings attributable to infection. Persisting fever without other signs or symptoms will not be interpreted as progressing infection
• Patients who receive corticosteroids at ≥ 0.5mg/kg prednisone or equivalent.
• Patients who received anti-thymocyte globulin (ATG, Alemtuzumab (Campath), or other T-Cell immunosupressive monoclonal antibodies in the last 28 days.
• Patients who received methotrexate, or other antimetabolite-type immunosuppressants that are toxic to proliferating T cells in the last 7 days.
• Patients who received extracorporeal photopheresis within the last 28 days.
• Patients who received checkpoint inhibitor agents (e.g., nivolumab, pembrolizumab, ipilimumab) within 3 drug half-lives of the most recent dose to the infusion of R-MVST.
• Received donor lymphocyte infusion in last 28 days.
• Evidence of GVHD ≥ grade 2
• Evidence of biopsy-proven acute rejection in SOT recipients
• Active and uncontrolled relapse of malignancy
• Patients who are pregnant, or breastfeeding.
• Female of childbearing potential, or male with a female partner of childbearing potential, unwilling to use a highly effective method of contraception.
• Uncontrolled intercurrent illness including, but not limited to symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, or psychiatric illness/social situations that would limit compliance with study requirements.
• Patients who have received investigational (IND) product within 14 days of infusion of the the R-MVST cells.

Donor inclusion and exclusion criteria will be followed as per the most current BMT SOP (Donor selection, Donor evaluation and Donor Deferral).