Inclusion Criteria for Procurement:  Diagnosis of refractory or recurrent B cell Acute Lymphoblastic Leukemia (B-ALL) with expression of CD19, CD20 and/or CD22 Projected age greater than or equal to 12 months at the time of TriCAR T cell infusion Life expectancy of greater than or equal to 8 weeks Weight greater than or equal to 10 kg Subjects greater than or equal to 18 years of age must have the ability to give informed consent according to applicable regulatory and local institutional requirements. Legal guardian's consent must be obtained for subjects < 18 years of age. Assent will be obtained from pediatric subjects and according to applicable regulatory and local institutional requirements. Adults with cognitive impairment who are unable to consent and those with Down's syndrome are also eligible for this protocol with consent/assent according to applicable regulatory and local institutional requirements.  Unless a subject has a previously obtained apheresis/phlebotomy product that is acceptable and available for manufacturing of CAR-T cells, the subject must discontinue all anti-cancer agents and, in the opinion of the investigator, have recovered from significant acute toxic effects of:  Chemotherapy and biologic agents: All chemotherapy and biologic therapy not specifically mentioned below must be discontinued greater than or equal to 7 days prior to collection, with the exception of intrathecal chemotherapy and maintenance chemotherapy but greater than or equal to 72 hours prior to collection (for the subset of subjects who relapse during maintenance); both of which may be administered at any point pre-study or upon enrollment Steroid use: All systemic corticosteroid therapy (unless physiologic replacement dosing of less than or equal to 12mg/m2/day hydrocortisone or equivalent) must be discontinued greater than or equal to 7 days prior to collection Tyrosine Kinase Inhibitor (TKI) use: All TKIs must be discontinued greater than or equal to 3 days prior to collection Hydroxyurea: must be discontinued greater than or equal to 1 day prior to collection Prior CAR-T cell therapy: must be at least 30 days from most recent CAR-T cell infusion prior to collection Immunotherapy directed at leukemia: No antibodies within three (3) half-lives prior to collection (or within 4 weeks) whichever is shorter. This includes Antithymocyte globulin (ATG) formulations. Anti T-cell Antibodies, Alemtuzumab: must be discontinued greater than or equal to 8 weeks prior to collection Absolute Lymphocyte count (ALC) greater than or equal to 100 cells/μL Subject willing to participate in long-term follow-up for up to 15 years if enrolled in the study  Inclusion Criteria for T-cell Therapy  Age greater than or equal to 12 months Weight greater than or equal 10.0 Kg Life expectancy of greater than or equal 6 weeks  B-ALL with no prior history of allo-HCT with one of the following:  Second or subsequent marrow relapse with or without extramedullary disease First marrow relapse at the end of re-induction with marrow having greater than or equal 0.01% blasts by morphology &/or flow cytometry with or without extramedullary disease Primary refractory disease defined by having greater than or equal 5% blasts in the marrow by morphology and/or minimal residual (MRD) testing after 2 or more separate induction regimens Subject has an indication for allo-HCT but deemed ineligible (including subjects who have persistent MRD prior to allo-HCT) CD19(+) or CD19(-) relapse or refractory ALL after infusion of CD19-targeted immunotherapy including CAR-T cells or blinatumomab (CD20 or CD22 expression is required for CD19- B-ALL).  Or B-ALL recurrent after allo-HCT defined as having greater than or equal 0.01% marrow disease  Available transduced T-cells with greater than or equal 15% expression of CD19, CD20 or CD22 CAR by flow cytometry.  Prohibited medications:  Cranial radiation therapy (inclusive of TBI) greater than or equal to 4 weeks Cytotoxic chemotherapy greater than or equal to 2 days Tyrosine Kinase Inhibitors greater than or equal to 7 days Total Bilirubin: less than or equal to 3X upper limit of normal (ULN) for age OR conjugated bilirubin less than or equal to 2mg/dl, except in subjects with Gilbert's syndrome where a total bilirubin level of up to 5.3 mg/dL will be acceptable ALT less than or equal to 5 times upper limit of normal Adequate renal function defined as serum creatinine that is less than or equal to maximum based on age/gender (as per table below) or Creatinine clearance or GFR (as measured or estimated by Cockcroft Gaultor Schwartz) greater than or equal 50 mL/min/1.73m2 Pulse oximetry of greater than or equal 90% on room air Left ventricular fractional shortening (LVFS) greater than or equal 28% confirmed by echocardiogram or left ventricular ejection fraction (LVEF) greater than or equal 45% confirmed by echocardiogram (MUGA or MRI heart may replace echocardiogram). Lansky score of greater than or equal 50% (age greater than or equal 1 and < 16 years) or Karnofsky score of greater than or equal 50% (age greater than or equal 16 years) Donor lymphocyte infusions (DLI) completed > 6 weeks prior to CAR-T cell infusion Subjects of childbearing/fathering potential must agree to use highly effective contraception from the time of initial T cell infusion through 12 months following the last T cell infusion Subjects > 18 years of age must have the ability to give informed consent according to applicable regulatory and local institutional requirements. Legal guardian's consent must be obtained for subjects < 18 years of age. Assent will be obtained from pediatric subjects and according to applicable regulatory and local institutional requirements. Adults with cognitive impairment who are unable to consent and those with Down's Syndrome are also eligible for this protocol with consent/assent according to applicable regulatory and local institutional requirements.  Exclusion Criteria for Procurement:  Active malignancy other than disease under study CNS 3 involvement with leukemia that, in the opinion of the investigator, cannot be controlled (to CNS 2 or 1) during the interval between enrollment and CAR-T cell infusion If history of allogeneic hematopoietic cell transplant (allo-HCT): active GVHD, or receiving immunosuppressive therapy for treatment or prevention of GVHD within 4 weeks prior to enrollment  Presence of active severe infection, defined as:  positive blood culture within 48 hours of collection, OR fever above 38.2° C, AND clinical signs of infection within 48 hours of collection (unless fever is attributed to leukemia) active viral infections including infection with HIV, hepatitis B, hepatitis C or HTLV Primary immunodeficiency syndrome Pregnant or breastfeeding Presence of any condition that, in the opinion of the investigator, would prohibit the subject from undergoing treatment under this protocol History of symptomatic CNS pathology or ongoing symptomatic CNS pathology requiring medical intervention, including paresis, aphasia, cerebrovascular ischemia/hemorrhage, severe brain injury, dementia, cerebellar disease, organic brain syndrome, psychosis, coordination or movement disorder (subjects with non-febrile seizure disorder controlled on anti-epileptic medication and without seizure activity within 3 months are eligible)  Exclusion Criteria for T-cell Therapy  Pregnant or lactating Presence of any condition that, in the opinion of the PI or designee, would prevent the patient from undergoing protocol-based therapy. Active CNS involvement by ALL, defined as CNS-3 or symptomatic CNS 2 (Refer to appendix III for definitions of CNS status). Note: Asymptomatic CNS2 or patients with history of CNS disease that has been effectively treated will be eligible. Patients that have a significant neurologic deterioration will be not be eligible for T cell infusion until alternate therapies result in neurological stabilization.