Viral Specific T-Lymphocytes by Cytokine Capture System (CCS) to Treat Infection With Adenovirus, Cytomegalovirus or Epstein-Barr Virus After Hematopoietic Cell Transplantation or Solid Organ Transplantation and in Patients With Compromised Immunity

Patient Inclusion Criteria 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 65 years old, inclusive, at the time of informed consent. 3. Prior allogeneic hematopoietic stem cell transplant, AND/OR prior solid organ transplant (liver, kidney, lung and/or heart, intestinal, pancreatic, and/or multivisceral), AND/OR diagnosis of primary immunodeficiency AND/OR current/recent administration of immunosuppressive therapy for cancer or autoimmune disease. 4. If receiving steroids, must be able to taper dose 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 or post-menopausal. 6. Diagnosis of Adenovirus, CMV, or EBV infection, persistent despite standard therapy. A. Adenovirus Infection or Disease (at minimum, one of the below sub-criteria must be met): 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) 2. Refractory adenoviremia: defined as DNAemia ≥1000 copies/mL or \<1 log decrease after at least 2 weeks of appropriate antiviral therapy (i.e. cidofovir, brincidofovir, or other available pharmacological agents) 3. Intolerance of or contraindication to antiviral medications. B. CMV Infection or Disease (at minimum, one of the below sub-criteria must be met): 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) 2. Refractory CMV viremia: defined as the continued presence of DNAemia, with ≥1,000 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) 3. Intolerance of or contraindication to antiviral medications. C. EBV Infection or Disease (at minimum, one of the below sub-criteria must be met): 1. EBV DNAemia ≥1000 IU/mL, persistent despite 2 doses of rituximab, 2. Biopsy proven lymphoma or lymphoproliferative disease with EBV genomes detected in tumor cells by immunocytochemistry (i.e. EBER positive) or in situ PCR, 3. Clinical or imaging findings consistent with EBV lymphoma or lymphoproliferation with current or recent elevated EBV viral load in peripheral blood in a patient where biopsy is deemed too high risk, 4. Failure of antiviral therapy, as determined by one of the two bullets below after two weeks of anti-CD20 targeted therapy such as rituximab, i. There was an increase or less than 50% response at sites of lymphoma disease or lymphoproliferation. ii. There was a rise or a fall of less than 50% in EBV viral load in peripheral blood. 5. Intolerance or contraindication to rituximab. Patient Exclusion Criteria: 1. Received ATG or Alemtuzumab within 21 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 planned 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 (other than EBV+ post-transplant lymphoproliferative disorder or lymphoma). 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, or concomitant medications, 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. Donor Inclusion Criteria 1. Age ≥ 12\* 2. Able to understand and sign the consent/assent to the procedure 3. Partial (2/6 or more) HLA match to the recipient 4. A pediatric donor could be selected as a donor only if a suitable adult donor is not available (as attested by the research team) or is ineligible according to FACT requirements. For pediatric donors: * Related to the recipient * Apheresis does not need a blood prime before the procedure * Adequate peripheral venous access * Explicit evaluation of the donors' willingness to donate cells, as attested by the research team * Must have understanding that they are helping their ill relative, as attested by the research team * Will gain emotional/psychological benefit from their ability to help and want to donate for a relative, as attested by the research team * Inclusion of minor donors that are not relatives of the recipient will need to be evaluated on a case-by-case basis for the IRB to evaluate the potential benefit to these participants (whether they will receive an emotional and psychological boost from helping the recipient) \*If the only suitable donor is less than 12 years old, a single patient exception to this inclusion criteria will be submitted and approved by the IRB before obtaining the donor's assent and their LAR consent. Donor Exclusion Criteria 1. Donor is pregnant 2. Donor is HIV positive 3. Donor is positive for hepatitis B and/or hepatitis C 4. Deemed to be a high-risk donor based on responses to donor risk questionnaire 5. Deemed high risk due to preexisting medical condition or abnormal lab results