CAR-20/19-T Cells in Patients With Relapsed/Refractory B Cell ALL

Inclusion Criteria: 1. Diagnosis of B-cell ALL: During the safety phase patients enrolled will be age ≥ 18 to age ≤ 70. Once those three patients have passed the 28-day waiting period post infusion, clearance from the FDA will be obtained to enroll pediatric patients. After FDA clearance -Patients must be aged ≥ 1 year and ≤ 70 years. All patients will have relapsed, refractory disease and no available curative options that meet clinical criteria to initiate treatment. 2. Relapsed or refractory B cell ALL defined as one of the following: 1. Primary refractory disease. 2. Relapsed or refractory disease after two or more lines of systemic therapy. 3. Relapsed or refractory disease after allogeneic transplant provided subject is at least 100 days from stem cell transplant at the time of enrollment and off of immunosuppressive medications for at least four weeks prior to enrollment. 3. Measurable or evaluable disease at the time of enrollment, which may include any evidence of disease including minimal residual disease detected by flow cytometry, or cytogenetics, or morphological disease in the bone marrow. 4. Subjects with B-cell ALL must have either CD19 or CD20 positive disease on their most recent bone marrow performed. A minimum of 5% CD19 or CD20 positivity on prior biopsy or bone marrow aspiration (BMA) is required. 5. Subjects with Ph+ ALL are eligible if they have relapsed or refractory disease and have failed at least two tyrosine kinase inhibitors. 6. Absolute cluster of differentiation 3 (CD3)+ T cell count ≥100/mm\^3. a. Subjects who receive chemotherapy and/or steroids after CD3+ T-cell count, but before apheresis, will require this test to be repeated. 7. Lumbar puncture with CSF analysis by cytology with no evidence of disease. 8. Karnofsky/Lansky performance score ≥70. 9. Normal Baseline Neurological Evaluation: Mini-Mental Status Exam Score 24-30. 10. Adequate hepatic function, defined as aspartate aminotransferase (AST) and alanine transaminase (ALT) \<3 x upper limit of normal (ULN); serum bilirubin and alkaline phosphatase \<2 x ULN, or considered not clinically significant as per the clinical PIs discretion (e.g. Gilbert's or indirect hyperbilirubinemia) or felt to be due to underlying disease. 11. Adequate renal function defined as creatinine clearance or radioisotope glomerular filtration rate (GFR) \> 70 ml/min/1.73 M2 12. Able to give informed consent if \> 18 years, or with a legal guardian capable of giving informed consent if \< 18 years. 13. Agree to practice birth control during the study. 14. Cardiac ejection fraction ≥ 50%, no evidence of pericardial effusion as determined by an echocardiogram (ECHO). 15. No clinically significant arrhythmias. 16. Adequate pulmonary function as indicated by room air oxygen saturation of ≥92% and no clinically significant pleural effusion. 17. Expected survival \>12 weeks. 18. Negative urine or serum pregnancy test in females of child bearing potential at study entry and again within 48 hours' prior to lymphodepleting chemotherapy. 19. Subjects with prior CD19 or CD20 therapy (e.g. blinatumomab, CART19, rituximab) treatment require repeat BMA post-CD19 or CD20 therapy treatment that demonstrates CD19 or CD20 positive disease. 20. Meet criteria for regarding fertility and contraception. Due to the high-risk level of this study, while enrolled, all subjects must agree not to participate in a conception process (e.g., active attempt to become pregnant or to impregnate, sperm donation, in vitro fertilization). Additionally, if participating in sexual activity that could lead to pregnancy, the study subject must agree to use reliable and double barrier methods of contraception during the follow-up period of the protocol. * Condoms (male or female) with or without a spermicidal agent. * Diaphragm or cervical cap with spermicide. * Intrauterine device (IUD). * Hormonal-based contraception. 21. Central line access will be required for CAR-20/19-T cell infusion. Exclusion Criteria: 1. Positive beta-human chorionic gonadotropin (HCG) in female of childbearing potential. 2. Subjects with known systemic allergy to bovine or murine products. 3. Confirmed active human immunodeficiency virus (HIV), Hepatitis B or C infection. A history of hepatitis B or hepatitis C is permitted if the viral load is undetectable per quantitative polymerase chain reaction (PCR) and/or nucleic acid testing. 4. History of significant autoimmune disease OR active, uncontrolled autoimmune phenomenon: such as systemic lupus erythematous, Wegner's glomerulonephritis, autoimmune hemolytic anemia, idiopathic thrombocytopenic purpura (AIHA, ITP) requiring steroid therapy defined as \>20 mg of prednisone. 5. Presence of ≥ grade 3 non-hematologic toxicities as per CTCAE version 5 from any previous treatment unless it is felt to be due to underlying disease. 6. Concurrent use of investigational therapeutic agents or enrollment on another therapeutic clinical trial at any institution. 7. Refusal to participate in the long-term follow-up protocol. 8. Central nervous system (CNS) Abnormalities: 1. Subjects with prior CNS disease that has been effectively treated will be eligible providing treatment was \> four weeks before enrollment and a remission documented within four weeks of planned CAR-T cell infusion. Subjects will be excluded if they have any signs of neurotoxicity at baseline or evidence of chloroma or leukemic infiltrates on MRI. 2. Presence of CNS-3 disease defined as detectable cerebrospinal blast cells in a sample of CSF with ≥ 5 white blood cells (WBCs) per mm\^3 with or without neurological changes, and presence of CNS-2 disease defined as detectable cerebrospinal blast cells in a sample of CSF with \< 5 WBCs per mm\^3 with neurological changes will be excluded. Note: Subjects with CNS-1 (no detectable leukemia in the CSF) and those with CNS-2 without clinically evident neurological changes are eligible to participate in the study. History or presence of any CNS disorder, such as a seizure disorder, cerebrovascular ischemia/hemorrhage, dementia, cerebellar disease, any autoimmune disease with CNS involvement, posterior reversible encephalopathy syndrome (PRES), or cerebral edema are excluded. 9. Previous recipients of allogeneic hematopoietic stem cell transplantation (AHCT) are excluded if they are \<100 days' post-transplant, have evidence of active graft-versus-host-disease (GVHD) of any grade, or are currently on immunosuppression. 10. Anti-CD20 antibody treatment within four weeks of cell infusion. 11. Anti-CD19 antibody treatment within four weeks of cell infusion. 12. Cytotoxic chemotherapy other than lymphodepletion within 14 days of CAR-T cell infusion. 13. Cytotoxic chemotherapy treatment within 14 days or steroid treatment (other than replacement dose steroids) within seven days prior to apheresis collection for CAR-T cells. Tyrosine kinase inhibitors (TKIs) must be held for five half lives or seven days whichever is shorter prior to enrollment. 14. Subjects post solid organ transplant who develop high grade lymphomas or leukemias. 15. Concurrent active malignancy (exceptions: treated solid malignancy in \> five years' remission, treated basal or squamous cell carcinomas of the skin). 16. History of concomitant genetic syndrome associated with bone marrow failure such as Fanconi anemia, Kostmann syndrome, or Shwachman-Diamond syndrome 17. No rapidly escalating ALL disease (please discuss with the principal investigator).