Haplo-identical Viral-Specific T-cells for Treatment of Cytomegalovirus and Adenovirus Infections After Hematopoietic Cell Transplantation

Inclusion Criteria for Patients: * Patients who have undergone haploidentical HCT or a matched-sibling/matched-unrelated donor HCT, and have CMV and/or ADV detected by PCR in the peripheral blood refractory to antiviral therapy per institutional BMTCT SOP 20.05. * Definition of "refractory" viremia is persistent positive CMV or ADV viremia after 14 days of treatment per institutional SOP, or an increasing copy number (≥1 log) after 7 days of treatment. * Patients have no suspected or confirmed GVHD. * Availability of haploidentical donor for isolation of virus-specific T-cells. * Have not received a Donor Lymphocyte Infusion in the past 4 weeks. * Female patients of childbearing age must have a negative pregnancy test. * Subject, parent, or guardian are capable of giving signed informed consent. * Patients must have a shortening fraction \>26% or left ventricular ejection fraction \>40%. * Patients must have a bilirubin less than or equal to 2.5mg/dL and alanine aminotransferase (ALT) less than or equal to 5 times the upper limit of normal. * Patients must have an estimated glomerular filtration rate (GFR) greater than 60mL/min/1.73m2 (may use estimated GFR that is auto calculated in the EHR). * Patients must be free of severe infection which upon determination of the principal investigator precludes therapy with VST. * Patients must have FVC \>50% predicted or able to maintain pulse oximetry saturation \> 92% on room air. * Gut diarrhea \<1 liter/day (adults) or \<20mL/kg/day (children) or if unable to quantify, then occurrence of 4 stools per day above baseline. * Patients must have engrafted with an ANC \>500 cells/mm3 for 3 consecutive days. Inclusion criteria for donors * Age ≥18 years. * At least single haplotype matched (≥3/6) family member. * Donor will be identical to the stem cell donor (Cohort A) or different from the stem cell donor (Cohort B). * HIV negative. * For females of childbearing age: Not pregnant as confirmed by negative serum or urine pregnancy test within 14 days prior to enrollment AND not lactating with intent to breastfeed. * Regarding donation eligibility, is identified as either having completed the process of donor eligibility determination as outlined in 21CFR 1271 and agency guidance or does not meet 21CFR 1271 eligibility requirements but has a declaration of urgent medical need completed by the principal investigator or physician sub-investigator per 21CFR. * Identified recipient with CMV and/or ADV reactivation post-HCT. Exclusion Criteria for Patients: * Active GVHD. * Pregnancy. * Inability to provide consent. * Need for vasopressor or ventilatory support Patients receiving steroids \>0.5 mg/kg prednisone equivalent at the time of VST infusion * Donor Lymphocyte Infusion within 4 weeks prior to VST infusion. * Receipt of Thymoglobulin or Alemtuzumab within 30 days of VST infusion. * Other severe uncontrolled concurrent infections (i.e. bacterial or fungal) that are not yet controlled on antimicrobial therapies.