The purpose of this phase I study is to test the safety of different dose levels of specially prepared cells collected from the patient called "modified T cells". The investigators want to find a safe dose of modified T cells for patients with this type of cancer that has progressed after standard therapy. The investigators also want to find out what effects these modified T cells have on the patient and the cancer.

For patients who were treated, had progression of disease and were removed from study, duplicate enrollment is permitted if it is determined the patients could receive a benefit. If the patients meet all eligibility criteria, they can be enrolled onto study a second time as a new accrual, and receive treatment in a higher dose level cohort.

Inclusion Criteria:

• Patients must have CD19+ B cell malignancy with relapsed or refractory disease, defined as below:

Patients with CLL:

• Refractory to or relapsed after at least 2 prior chemo or chemoimmunotherapy (e.g. FCR, BR) requiring further treatment
• Refactory to or relapsed after at least 1 prior biologic agent (e.g. Ibrutinib, idelalisib, venetoclax, except a single agent anti-CD20 monoclonal antibody) requiring further treatment

Patients with iNHL (FZ, MZL, WM):

• Refractory or relapsed after at least 2 lines of chemoimmunotherapy (including at least one course of anti-CD20 antibody)
• Refractory or relapsed after at least 1 prior biologic agent (e.g. lenalidomide, ibrutinib, idelalisib)
• Patients must have measurable disease (for WM patients, measureable disease is demonstrable monoclonal paraprotein and bone marrow involvement)

Patients with DLBCL, Transformed B cell lymphoma, or High grade B cell lymphoma:

• Refractory to or relapsed after 1 or more prior chemoimmunotherapies with at least one containing an anthracycline and CD20 directed therapy
• Transplant ineligible
• Biopsy proven relapsed disease

Patients with ALL, CML in lymphoid blast crisis or Burkitt's lymphoma:

• Refractory to at least 1 prior induction chemotherapy

• Relapsed after at least 1 prior multiagent systemic chemotherapy that included induction and consolidation

• Patients with Philadelphia chromosome-positive ALL must have failed a second generation tyrosine kinase inhibitor

  • Age ≥ 18 years of age
  • Creatinine ≤2.0 mg/100 ml, direct bilirubin ≤2.0 mg/100 ml, AST and ALT ≤3.0x upper limit of normal (ULN)
  • Adequate pulmonary function as assessed by ≥92% oxygen saturation on room air by pulse oximetry

Exclusion Criteria:

• Karnofsky performance status <70

• Pregnant or lactating women. Women and men of childbearing age should use effective contraception while on this study and continue for 1 year after all treatment is finished

• Impaired cardiac function (LVEF <40%) as assessed by ECHO or MUGA scan

• Patients with active known autoimmune disease are ineligible

• Patients with following cardiac conditions will be excluded:

  • New York Heart Association (NYHA) stage III or IV congestive heart failure
  • Myocardial infarction \