Activated T-Cells Expressing 2nd or 3rd Generation CD19-Specific CAR, Advanced B-Cell NHL, ALL, and CLL (SAGAN)

Inclusion Criteria: PROCUREMENT Referred patients (or respective donors) will initially be consented for procurement of blood for generation of the transduced ATL. Eligibility criteria at this stage include: * Diagnosis of recurrent B-cell lymphoma or leukemia (ALL or CLL), or newly diagnosed patients unable to receive or complete standard therapy OR diagnosis of relapsed/refractory aggressive B-cell lymphoma with a treatment plan that will include high dose therapy and autologous stem cell transplantation. * CD19-positive tumor (result can be pending at this time). * Age \<= 75 years. The first 3 patients treated on the study should be adults (\>= 18 years). * Hgb greater than or equal to 7.0 (can be a transfused value) * If pheresis required to collect blood: * Creatinine \< 1.5 x upper limit normal * AST \<1.5 × upper limit normal * PT and APTT \<1.5 × upper limit normal * Informed consent explained to, understood by and signed by patient/guardian (and donor, where applicable). Patient/guardian given copy of informed consent. TREATMENT * Diagnosis of recurrent B-cell lymphoma leukemia (ALL or CLL), or newly diagnosed patients unable to receive or complete standard therapy OR diagnosis of relapsed/refractory aggressive B-cell lymphoma with a treatment plan that will include high dose therapy and autologous stem cell transplantation. * CD19-positive tumor. * Age \<= 75 years. The first 3 patients treated on the study should be adults (\>= 18 years). * Bilirubin less than 3 times the upper limit of normal. * AST less than 5 times the upper limit of normal. * Estimated GFR \> 50 mL/min * Pulse oximetry of \> 90% on room air * Karnofsky or Lansky score of \> 60%. * Recovered from acute toxic effects of prior chemotherapy at least one week before entering this study. PD1/PDL1 inhibitors will be allowed if medically indicated. * Available autologous or syngeneic activated peripheral blood T cell products (CD28ζ and CD28/CD137ζ) with more than or equal to 15% expression of CD19.CAR determined by flow cytometry. * Life expectancy of greater than 12 weeks. * Sexually active patients must be willing to utilize one of the more effective birth control methods during the study and for 6 months after the study is concluded. The male partner should use a condom. * Patients or legal guardians must sign an informed consent indicating that they are aware this is a research study and have been told of its possible benefits and toxic side effects. Patients or their guardians will be given a copy of the consent form. Exclusion Criteria: PROCUREMENT * Active infection requiring antibiotics. * No history of other cancer (except non-melanoma skin cancer or in situ breast cancer or cervix cancer) unless the tumor was successfully treated with curative intent at least 2 years before trial entry. TREATMENT * Currently receiving any investigational agents or received any tumor vaccines within the previous 6 weeks. (Note treatment with PD1/PDL1 inhibitors is allowed.) * History of hypersensitivity reactions to murine protein-containing products. * Pregnant or lactating. * Tumor in a location where enlargement could cause airway obstruction. * Active infection with HIV or HTLV.