Inclusion Criteria:

   - In view of the principal investigator (PI) and the primary oncologist, there must be
   no available alternative curative therapies and subjects must be either ineligible for
   allogeneic stem cell transplant (SCT), have refused SCT, recurred after SCT, or have
   disease activity that prohibits SCT at the time of enrollment

      - Chemotherapy refractory disease in subjects with B-ALL is defined as progression
      or stable disease after two lines of standard therapies (two induction regimens),
      or relapsed disease after 2 established induction regimens

      - Subjects with persistent or relapsed minimal residual disease (MRD)/next
      generation sequencing (NGS) relapse require verification of relapse on two
      occasions at least 4 weeks apart

      - Subjects with Philadelphia chromosome positive acute lymphoblastic leukemia (Ph+
      ALL) subjects are eligible if they progressed, had stable disease or relapsed
      after two lines of therapy, including tyrosine kinase inhibitors (TKIs)

      - Subjects with lymphoma must have progressed, had stable disease (SD), or recurred
      after two standard treatment regimens

   - CD19 expression is required and must be detected on greater than 50% of the malignant
   cells by immunohistochemistry or >= 90% by flow cytometry; the choice of whether to
   use flow cytometry or immunohistochemistry will be determined by what is the most
   easily available tissue sample in each subject; in general, immunohistochemistry will
   be used for lymph node biopsies, flow cytometry will be used for peripheral blood and
   bone marrow samples

   - Subjects who have undergone autologous SCT will be eligible if all other eligibility
   criteria are met; subjects who have undergone allogeneic SCT will be eligible if, in
   addition to meeting other eligibility criteria, they are at least 100 days
   post-transplant, they have no evidence of active graft versus host disease (GVHD) and
   have been without immunosuppressive agents for at least 30 days

   - For patients with lymphoma, evaluable or measurable disease according to the revised
   International Working Group (IWG) Response Criteria for Malignant Lymphoma must be
   present; lesions that have been previously irradiated will be considered measurable
   only if progression has been documented following completion of radiation therapy

   - At least 2 weeks or 5 half-lives, whichever is shorter, must have elapsed since any
   prior systemic therapy at the time the subject is planned for leukapheresis, except
   for systemic inhibitory/stimulatory immune checkpoint therapy, which requires 5
   half-lives

      - Exceptions:

         - There is no time restriction with regard to prior intrathecal chemotherapy
         provided there is complete recovery from any acute toxic effects of such

         - Subjects receiving hydroxyurea may be enrolled provided there has been no
         increase in dose for at least 2 weeks prior to starting apheresis

         - Subjects who are on standard ALL maintenance type chemotherapy (vincristine,
         6-mercaptopurine or oral methotrexate) may be enrolled provided that
         chemotherapy is discontinued at least 1 week prior to apheresis

         - Subjects receiving steroid therapy at physiologic replacement doses only are
         allowed provided there has been no increase in dose for at least 2 weeks
         prior to starting apheresis

         - For radiation therapy: radiation therapy must have been completed at least 3
         weeks prior to enrollment, with the exception that there is no time
         restriction if the volume of bone marrow treated is less than 10% and also
         the subject has measurable/evaluable disease outside the radiation port

   - Toxicities due to prior therapy must be stable and recovered to =< grade 1 (except for
   clinically non-significant toxicities such as alopecia)

   - NOTE: the first subject in the first dose cohort must be >= 18 years of age if an
   adult has not been treated at that dose cohort on the companion Stanford protocol
   "Phase 1 Dose Escalation Study of CD19/CD22 Chimeric Antigen Receptor (CAR) T Cells in
   Adults with Recurrent or Refractory B Cell Malignancies" and undergone safety
   evaluation at day 28 without evidence of dose limiting toxicity (DLT)

   - Subjects > 10 years of age: Karnofsky >= 50%; Subjects =< 10 years of age: Lansky
   scale >= 50%

   - Absolute neutrophil count (ANC) >= 750/uL if these cytopenias are not judged by the
   investigator to be due to underlying disease (i.e. potentially reversible with
   anti-neoplastic therapy); a subject will not be excluded because of pancytopenia >=
   grade 3 if it is due to disease, based on the results of bone marrow studies

   - Platelet count >= 50,000/uL if these cytopenias are not judged by the investigator to
   be due to underlying disease (i.e. potentially reversible with anti-neoplastic
   therapy); a subject will not be excluded because of pancytopenia >= grade 3 if it is
   due to disease, based on the results of bone marrow studies

   - Absolute lymphocyte count >= 150/uL

   - Serum alanine aminotransferase (ALT)/aspartate aminotransferase (AST) =< 2.5 upper
   limit of normal (ULN)

   - Total bilirubin =< 1.5 mg/dl, except in subjects with Gilbert's syndrome

   - Cardiac ejection fraction >= 45%, no evidence of physiologically significant
   pericardial effusion as determined by an echocardiogram (ECHO), and no clinically
   significant electrocardiogram (ECG) findings

   - No clinically significant pleural effusion

   - Baseline oxygen saturation > 92% on room air

   - Creatinine: within age adjusted normal institutional limits:

      - Age =< 5 years - maximum serum creatinine 0.8 mg/dL

      - Age 5 years < age =< 10 years - maximum serum creatinine 1.0 mg/dL

      - Age > 10 years - maximum serum creatinine 1.2 mg/dL OR

   - Creatinine clearance >= 60 mL/min/1.73 m^2 (as estimated by Cockcroft Gault equation)
   for subjects with creatinine levels above institutional normal

   - Central nervous system (CNS) status

      - Subjects with ALL

         - Subjects with the following CNS status are eligible only in the absence of
         neurologic symptoms suggestive of CNS leukemia, such as cranial nerve palsy:

            - CNS 1, defined as absence of blasts in cerebral spinal fluid (CSF) on
            cytospin preparation, regardless of the number of white blood cells
            (WBCs)

            - CNS 2, defined as presence of < 5/uL WBCs in CSF and cytospin positive
            for blasts, or > 5/uL WBCs but negative by Steinherz/Bleyer algorithm:

               - CNS 2a: < 10/uL red blood cells (RBCs); < 5/uL WBCs and cytospin
               positive for blasts

               - CNS 2b: >= 10/uL RBCs; < 5/uL WBCs and cytospin positive for
               blasts

               - CNS 2c: >= 10/uL RBCs; >= 5/uL WBCs and cytospin positive for
               blasts but negative by Steinherz/Bleyer algorithm

      - Subjects with lymphoma

         - Subjects must have no signs or symptoms of CNS disease or detectable
         evidence of CNS disease on magnetic resonance imaging (MRI) at the time of
         screening; subjects who have been previously treated for CNS disease but
         have no evidence of disease at screening are eligible

   - Females of childbearing potential must have a negative serum or urine pregnancy test
   (females who have undergone surgical sterilization or who have been postmenopausal for
   at least 2 years are not considered to be of childbearing potential)

   - Subjects 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 (4) months after
   receiving the preparative regimen; females of child-bearing potential must have a
   negative pregnancy test

   - Ability to give informed consent; all subjects >= 18 years of age must be able to give
   informed consent; for subjects < 18 years old their legal authorized representative
   (LAR) (i.e. parent or guardian) must give informed consent; pediatric subjects will be
   included in age appropriate discussion and verbal assent will be obtained for those >
   7 years of age, when appropriate

Exclusion Criteria:

   - Recurrent or refractory ALL limited to isolated testicular or isolated central nervous
   system (CNS) disease

   - Subjects with radiologically-detected CNS lymphoma or CNS 3 disease (presence of >=
   5/uL WBCs in CSF and cytospin positive for blasts [in the absence of a traumatic
   lumbar puncture] and/or clinical signs of CNS leukemia)

   - Hyperleukocytosis (>= 50,000 blasts/uL) or rapidly progressive disease that in the
   estimation of the investigator and sponsor would compromise ability to complete study
   therapy

   - History of malignancy other than non-melanoma skin cancer or carcinoma in situ (e.g.
   cervix, bladder, breast) unless disease free for at least 3 years

   - Presence of fungal, bacterial, viral, or other infection that is uncontrolled or
   requiring IV antimicrobials for management; simple urinary tract infection (UTI) and
   uncomplicated bacterial pharyngitis are permitted if responding to active treatment;
   ongoing infection with human immunodeficiency virus (HIV) or hepatitis B (hepatitis B
   surface antigen [HBsAg] positive) or hepatitis C virus (anti hepatitis C virus [HCV]
   positive); 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

   - CNS disorder such as cerebrovascular ischemia/hemorrhage, dementia, cerebellar
   disease, or autoimmune disease with CNS involvement

   - History of myocardial infarction, cardiac angioplasty or stenting, unstable angina, or
   other clinically significant cardiac disease within 12 months of enrollment, or have
   cardiac atrial or cardiac ventricular lymphoma involvement

   - Subjects receiving anticoagulation therapy

   - Any medical condition that in the judgement of the sponsor investigator is likely to
   interfere with assessment of safety or efficacy of study treatment

   - History of severe immediate hypersensitivity reaction to any of the agents used in
   this study

   - Women of child-bearing potential who are pregnant or breastfeeding; females who have
   undergone surgical sterilization or who have been postmenopausal for at least 2 years
   are not considered to be of childbearing potential

   - In the investigator's judgment, the subject is unlikely to complete all
   protocol-required study visits or procedures, including follow-up visits, or comply
   with the study requirements for participation

   - May not have primary immunodeficiency or history of systemic autoimmune disease (e.g.
   Crohns, rheumatoid arthritis, systemic lupus) resulting in end organ injury or
   requiring systemic immunosuppression/systemic disease modifying agents within the last
   2 years