Viral Specific T-Lymphocytes to Treat Adenovirus or CMV

1. Patient, parent, or legal guardian must have given written informed consent, according to FDA guidelines. For patients ≥ 7 years of age who are developmentally able, assent or affirmation will be obtained, if feasible. 2. Male or female, 1 month through 60 years old, inclusive, at the time of informed consent. 3. Prior allogeneic hematopoietic stem cell transplant (bone marrow, peripheral blood stem cells, single or double cord blood), OR prior solid organ transplant (liver, kidney, lung and/or heart, intestinal, or multivisceral), OR diagnosis of primary immunodeficiency OR current/recent administration of immunosuppressive therapy for cancer or autoimmune disease. 4. Clinical status, at time of consent, amendable to tapering of steroids to less than 1 mg/kg/day prednisone (or equivalent) prior to cellular infusion. 5. Negative pregnancy test for females ≥10 years old or who have reached menarche, unless surgically sterilized. 6. Diagnosis of Adenovirus or CMV infection, persistent despite standard therapy. A. Adenovirus Infection or Disease: 1. Active adenovirus infection: (i.e. gastroenteritis, pneumonia, hemorrhagic cystitis, hepatitis, pancreatitis, meningitis) defined as the demonstration of adenovirus by biopsy specimen from affected site(s) (by culture or histology), or the detection of adenovirus by culture, PCR or direct fluorescent antibody stain in fluid in the presence of worsening or persistent clinical or imaging findings despite at least 14 days of appropriate antiviral therapy (i.e. cidofovir, brincidofovir, or other available pharmacological agents) OR 2. Refractory adenoviremia: defined as DNAemia \>5000 copies/mL or \<1 log decrease after at least 2 weeks of appropriate antiviral therapy (i.e. cidofovir, brincidofovir, or other available pharmacological agents) OR 3. Intolerance of or contraindication to antiviral medications. B. CMV Infection or Disease: 1. Active CMV infection: (i.e. pneumonia, meningitis, retinitis, hepatitis, hemorrhagic cystitis, and/or gastroenteritis) defined as the demonstration of CMV by biopsy specimen from affected site(s) (by culture or histology) or the detection of CMV by culture, PCR or direct fluorescent antibody stain in fluid in the presence of worsening or persistent clinical or imaging findings despite at least 14 days of appropriate antiviral therapy (i.e. Foscarnet, ganciclovir, cidofovir, or other available pharmacological agents) OR 2. Refractory CMV viremia: defined as the continued presence of DNAemia, with ≥2,000 IU/mL or \<1 log decrease after at least 14 days of appropriate antiviral therapy (i.e. Foscarnet, ganciclovir, cidofovir, or other available pharmacological agents) OR 3. Intolerance of or contraindication to antiviral medications. 7. Donor Eligibility Criteria 1. 12 years of age or older 2. Able to understand and consent/assent to the procedure 3. Required hemoglobin of 11g/dL Exclusion Criteria: 1. Received ATG or Alemtuzumab within 28 days of viral-specific T cell infusion and a lack of evidence of T cell survival, defined by \<10 CD3+ T cells/uL (in unique situations, plasmapheresis may be considered). 2. Active acute GVHD grades II-IV. 3. Active severe chronic GVHD. 4. Received donor lymphocyte infusion, with the exception of a fraction of an umbilical cord blood, within 21 days of viral-specific T cell infusion. Subjects receiving a fraction of an umbilical cord blood within 21 days of the viral-specific T cell infusion will not be excluded. 5. Active and uncontrolled relapse of malignancy. 6. Anticipated initiation of new lymphotoxic therapy within 4 weeks of viral-specific T cell infusion. 7. Patients who are pregnant or lactating. 8. Past or current medical problems or findings from physical examination or laboratory testing that are not listed above, which, in the opinion of the investigator, may pose additional risks to participation in the study, may interfere with the participant's ability to comply with study requirements, or that may impact the quality or interpretation of the data obtained from the study. 9. Donor Exclusion Criteria 1. Patients who are pregnant 2. Patients who are HIV positive 3. Uncontrolled infection 4. Deemed high risk due to pre-existing medical condition