This study aims at evaluating the feasibility and safety of the administration of autologous T cells that have been modified through the introduction of a chimeric antigen receptor targeting the B-cell surface antigen CD19, following administration of lymphodepleting chemotherapy regimen, in children and adults with relapsed/refractory B-cell acute lymphoblastic leukemia (B- ALL) or aggressive B-cell Non-Hodgkin lymphoma (B-NHL). The phase II extension is aimed at testing the efficacy of the treatment at the optimal dose defined in the phase I. In addition, the investigators hypothesize that it is feasible to successfully manufacture CAR T cells to meet the established release criteria at a maximum target dose of 3.0 x 10^6 cells/kilogram recipient total body weight in this patient population using the Miltenyi CliniMACS Prodigy® closed transduction system.

Inclusion Criteria:

1. Diagnosis of CD19 expressing B-ALL or DLBCL or PML and one of the following:

   1. Patients in 1st relapse, with High-Risk (HR) features including: MLL- rearrangements, E2A/TCF3-PBX1, TCF3-HLF [t(17;19)], hypodiploidy (i.e., <44 chromosomes), TP53 alterations, early (i.e., <30 months from diagnosis)/very early (i.e., <18 months from diagnosis) isolated or combined bone marrow relapse
   2. MRD > 0.1% after either reinduction therapy or any course of consolidation for relapsed ALL
   3. Patients with DLBCL or PML in 1st or subsequent relapse, after at least one standard frontline chemotherapy

2. Age: 1 year - 25 years for Bcp-ALL and 1-35 years for B-NHL.

3. Voluntary informed consent is given. For subjects < 18 year-old their legal guardian must give informed consent. Pediatric subjects will be included in age-appropriate discussion and verbal assent will be obtained for those greater than or equal to 12 years of age, when appropriate.

4. Clinical performance status: Patients > 16 years of age: Karnofsky greater than or equal to 60%; Patients < 16 years of age: Lansky scale greater than or equal to 60%.

5. Patients of child-bearing or child-fathering potential must be willing to practice birth control from the time of enrollment on this study and for four months after receiving the preparative regimen.

6. Females of child-bearing potential must have a negative pregnancy test because of the potentially dangerous effects on the fetus.

Exclusion Criteria:

1. Pregnant or lactating women

2. Severe, uncontrolled active infections

3. HIV, or active HCV and/or HBV infection (detection of viral RNA/DNA in blood)

4. Life-expectancy < 6 weeks

5. Hepatic function: Inadequate liver function defined as total bilirubin > 4x upper limit of normal (ULN) or transaminase (ALT and AST) > 6 x ULN

6. Renal function: serum creatinine > 3x ULN for age.

7. Blood oxygen saturation < 90%.

8. Cardiac function: Left ventricular ejection fraction lower than 45% by ECHO.

9. Congestive heart failure, cardiac arrhythmia, psychiatric illness, or social situations that would limit compliance with study requirements or in the opinion of the PI would pose an unacceptable risk to the subject.

10. BM blasts > 50% pre-infusion.

11. Hyperleukocytosis (greater than or equal to 20,000 blasts/microliter) or rapidly progressive disease that in the evaluation of the investigator would compromise ability to complete study therapy

12. Presence of active, grade 2-4 acute or moderate-severe chronic GvHD

13. Recurrent or refractory ALL with testicular involvement

14. Concurrent or recent prior therapies, before infusion:

    1. Systemic steroids (at a dose > 2 mg/kg prednisone) in the 2 weeks before infusion. Recent or current use of inhaled/topical/non-absorbable steroids is not exclusionary.
    2. Systemic chemotherapy in the week preceding infusion.
    3. Anti-thymocyte globulin (ATG) in the 4 weeks preceding infusion.
    4. Immunosuppressive agents in the 1 week preceding infusion.
    5. Radiation therapy must have been completed at least 3 weeks prior to enrollment.
    6. Other anti-neoplastic investigational agents currently administered or within 30 days prior to infusion (i.e. start of protocol therapy);
    7. Exceptions:

    i. There is no time restriction with respect to prior intrathecal chemotherapy, provided that there is complete recovery from any acute toxic effects of such; ii. Patients who relapse while receiving standard ALL maintenance chemotherapy will not be required to have a waiting period before entry onto this study provided that they meet all other eligibility criteria; iii. Subjects receiving steroid therapy at physiologic replacement doses only are allowed provided that there has been no increase in dose for at least 2 weeks prior to starting apheresis;

15. Patient-derived CD19-CAR_Lenti production failure: vitality of the fresh product <80%, CD3+ cells <80%, CD3+ CAR+ cells <10%, non-sterility in IPC at day 5, endotoxin contamination (> 5 EU/ml) in IPC at day 5, mycoplasma contamination in IPC at day 5, failure of the visual inspection.