Patients with relapsed or refractory leukemia or lymphoma are often refractory to further chemotherapy. In this study, the investigators will attempt to use T cells obtained directly from the patient, which can be genetically engineered to express a chimeric antigen receptor (CAR). The CAR used in this study can recognize CD22, a protein expressed on the surface of leukemia and lymphoma cells. The phase 1 part of this study will determine the safety and appropriate dose level of these CAR T cells, and the phase 2 part of the study will determine how effective this CAR T cell therapy is. Both patients who have never had prior CAR T cell therapy and those who have had prior CAR T cell therapy may be eligible to participate in this study.

Inclusion Criteria:

• Male and female subjects aged ≤ 30 years. First 2 enrolled subjects: age ≥ 18 and ≤ 30 years
• Evidence of refractory or recurrent CD22+ leukemia or lymphoma
• Able to tolerate apheresis, or subject with sufficient existing apheresis product or T cells for manufacturing investigational product.
• Life expectancy ≥ 8 weeks
• Lansky or Karnofsky, as applicable, score ≥ 50
• Recovered from acute toxic effects of all prior chemotherapy, immunotherapy, and radiotherapy, if the subject does not have a previously obtained apheresis product that is acceptable and available for manufacturing of CAR T cells
• ≥ 7 days post last chemotherapy and biologic therapy, with the exception of intrathecal chemotherapy and maintenance chemotherapy
• ≥ 7 days post last corticosteroid therapy
• ≥ 3 days post Tyrosine Kinase Inhibitor (TKI) use
• ≥ 1 day post hydroxyurea
• 30 days post most recent CAR T cell infusion
• Adequate organ function
• Adequate laboratory values, including absolute lymphocyte count ≥ 100 cells/uL
• Subjects of childbearing or child-fathering potential must agree to use highly effective contraception from consent through 12 months following infusion of investigational product on trial
• Subject and/or legally authorized representative has signed the informed consent form for this study

Exclusion Criteria:

• Presence of active malignancy other than disease under study
• History of symptomatic CNS pathology or ongoing symptomatic CNS pathology
• CNS involvement of leukemia or lymphoma that is symptomatic and in the opinion of the investigator, cannot be controlled during the interval between enrollment and CAR T cell infusion
• Subjects with uniform expression of CD19 on their malignant cells who are eligible but have not attempted CD19 directed CAR T cell therapy
• For subjects having had a previous stem cell transplant: presence of active GVHD, or receiving immunosuppressive therapy for treatment or prevention of GVHD within 4 weeks prior to enrollment
• Presence of active severe infection,
• Presence of primary immunodeficiency syndrome
• Subject has received prior virotherapy
• Pregnant or breastfeeding
• Subject and/or legally authorized representative unwilling to provide consent/assent for participation in the 15-year follow-up period, required if CAR T cell therapy is administered
• Presence of any condition that, in the opinion of the investigator, would prohibit the subject from undergoing treatment under this protocol