Adoptive Cord Blood Immunotherapy for EBV, CMV, BKV and Adenovirus Reactivation/Infection or Prophylaxis

Inclusion Criteria: Inclusion Criteria at the Time of Procurement * Pediatric and adult patients (there are no lower and upper age limits for patients) with malignant or nonmalignant diseases who are candidates for transplant. * Patients must have a CB unit (or units) matched with the patient at 4, 5, or 6/6 HLA class I (serological) and II (molecular) antigens. The unit selected for CTL expansion must be either: 1. be cryopreserved in two fractions, with a minimum of 2.5x107 total nucleated cells (TNC) per kg pre-thaw in the fraction which will be used for the primary transplant. The remaining fraction will be used to generate the CTLs to give at day 30 or beyond as described below. OR 2. be cryopreserved with a cell dose that totals \> 3x10e7 TNC per kg pre thaw. On thaw, 20% of the total volume will be used for CTL manufacture to give at day 30 or beyond and the remaining 80% will be used for the primary transplant. 3. For recipients of double CBT, if possible both CB units should be cryopreserved in two fractions and T-cells will be made from both units if possible. Inclusion Criteria at the Time of CTL Infusion * Recipients of at least one unmanipulated cord blood unit as described above (i.e. from a HLA matched or mismatched unrelated donor) transplant at risk for or with CMV/Adenoviral/BKV and/or EBV infection or reactivation. * Lansky/Karnofsky scores ≥60 * Absolute neutrophil count (ANC) greater than 500/u * No evidence of GVHD \> Grade II at time of enrollment * Life expectancy \> 30 days * Absence of severe renal disease (Creatinine \< 3x normal for age) * Absence of severe hepatic disease. Direct bilirubin must be \< 3 mg/dl and AST \< 5x upper limit of normal * Patient must be at least 30 days post transplant to be eligible to receive CTL * Written informed consent and/or signed assent line from patient, parent or guardian Exclusion Criteria: Exclusion Criteria at the Time of Procurement * Pregnant or lactating * Patients with active central nervous system disease * Patients with Karnofsky performance status \<70% * Patients with grade 3 or 4 or primary myelofibrosis * Patients with suitable related donors Exclusion Criteria at the Time of CTL Infusion * Pregnant or lactating * Patient on Fi02 of \>60% * Unable to wean steroids to ≤0.5 mg/kg/day prednisone or equivalent * Patients with Grade 3 hyperbilirubinemia * Patients with other uncontrolled infections (except CMV and/or adenovirus and/or EBVemia and/or BK viruria/viremia). For bacterial infections, patients must be receiving definitive therapy and have no signs of progressing infection for 72 hours prior to enrollment. For fungal infections patients must be receiving definitive systemic anti-fungal therapy and have no signs of progressing infection for 1 week prior to enrollment. Progressing infection is defined as hemodynamic instability attributable to sepsis or new symptoms, worsening physical signs or radiographic findings attributable to infection. Persisting fever without other signs or symptoms will not be interpreted as progressing infection. * Patients with less than 50% donor chimerism in either peripheral blood or bone marrow or patients with relapse of original disease * Patients who have received investigational (IND) product within 28 days of screening for CTL infusion under this study