Donor Virus-Specific CMV or AdV CTL to Treat CMV or AdV Reactivation or Disease After Solid Organ or HCT

Inclusion Criteria: * Patients must have solid organ transplant or have received allogeneic hematopoietic stem cell transplant. * • Cohort A (CMV): Must have documented CMV disease or reactivation, as by: * Viremia as detected by quantitative polymerase chain reaction (PCR) (\> 500 IU/ml) in the peripheral blood requiring treatment OR * High risk for antiviral failure due to history of recurrent CMV reactivations or evidence of antiviral drug resistance, OR * Unable to tolerate antiviral drugs due to renal toxicity, bone marrow suppression, transfusion dependent anemia and thrombocytopenia or neutropenia requiring growth factor support or other related organ injury • Cohort B (AdV): Must have documented AdV infection or reactivation, as by: * Symptomatic subject with any detectable viral load in blood, OR * Symptomatic subject with qualitative AdV detection in compartment of current symptomatology, including stool, urine, and/or other specimens (bronchoalveolar lavage (BAL), nasal swab, CSF, etc.), irrespective of blood viral load, OR * New, persistent, and/or worsening AdV-related symptoms, signs, and/or markers of end organ compromise while receiving antiviral therapy (ie cidofovir), OR * Asymptomatic with a viral load \> 1000 copies/ml in peripheral blood, OR * Unable to tolerate antiviral treatment due to renal toxicity, bone marrow suppression, transfusion dependent anemia and thrombocytopenia or neutropenia requiring growth factor support or other related organ injury * Karnofsky (age \> 16 years) or Lansky performance score \> 70 (age \< 16) * Available seropositive haploidentical or matched donor who is without evidence of infection that would otherwise preclude donation * Negative pregnancy test in female patients if applicable (childbearing potential, has not received a full-intensity conditioning regimen * Written informed consent and/or signed assent line from patient, parent or guardian * DONOR * Human leukocyte antigen (HLA)-haploidentical or full-match to the patient as determined by institutional standards * Cohort A: CMV seropositive, defined as detection of serum CMV immunoglobulin G (IgG) * Cohort B: AdV seropositive, defined as detection of serum AdV IgG * Age 18 or over * Meet donor eligibility or suitability according to institutional standards. If the donor is deemed ineligible according to Foundation for the Accreditation of Cellular Therapy (FACT) standards, but is suitable for donation per institutional standards, the donor will be eligible for the protocol Exclusion Criteria: * Receipt of anti-thymocyte globulin (ATG), alemtuzumab, or other T-cell depleting agents within 21 days of screening for enrollment. * Receipt of \> 0.5mg/kg/day of prednisone or steroid equivalent at the time of enrollment. Stable GVHD is permitted as long as patients are on stable dose steroids of less than or equal to 0.5 mg/kg/day of prednisone or steroid equivalent. * Evidence of uncontrolled infection as follows: * Bacterial infections - patients must be receiving definitive therapy and have no signs of progressing infection for 72 hours prior to enrollment. * Fungal infections - patients must be receiving definitive systemic anti-fungal therapy and have no signs of progressing infection for 1 week prior to enrollment. * Patients with hemodynamic instability attributable to bacterial sepsis or new symptoms, worsening physical signs or radiographic findings attributable to concomitant bacterial or fungal infection are excluded. Patients who require ventilator support for CMV pneumonitis are not excluded. Persisting fever without other signs or symptoms will not be interpreted as progressing infection. * Receipt of donor lymphocyte infusion (DLI) within 28 days. * Patients with active acute graft versus host disease (GvHD) grades II-IV requiring \> 0.5 mg/kg/day of prednisone or steroid equivalent or T-cell depleting immunosuppression. * Acute graft rejection in solid organ transplantation requiring augmented immunosuppression with T-cell depleting agents or steroids as mentioned above. * Active and uncontrolled relapse of malignancy.