Additional Study Information: The purpose of this study is to test whether giving an experimental cell product can treat viral infection in patients who have conditions that cause the poor function of their immune system. The patients with infections caused by viruses such as cytomegalovirus (CMV), Epstein-Barr virus (EBV), adenovirus (ADV), BK and JC polyomaviruses (PyVs) virus may participate in this study. The cell product is called rapidly generated virus-specific T cells (R-MVST). R-MVST cells will be manufactured in our laboratory at Columbia University Irving Medical Center. This cell product has not been approved by the Food and Drug Administration (FDA).