CD19-directed CAR-T cell therapy has shown promising results for the treatment of relapsed or refractory B-cell malignancies; however, a subset of patients relapse due to the loss of CD19 in tumor cells. Dual Specificity CD19 and CD20 or CD22 CAR-T cells can recognize and kill the CD19 negative malignant cells through recognition of CD20 or CD22. This is a phase 1/2 study designed to determine the safety of the allogenic gene-edited dual specificity CD19 and CD20 or CD22 CAR-T cells and the feasibility of making enough to treat patients with relapsed or refractory hematological malignancies.

1. PRIMARY OBJECTIVES:

   1. To evaluate the feasibility and safety of universal dual specificity CD19 and CD20 or CD22 CAR-T cells in patients with relapsed or refractory leukemia and lymphoma.
   2. To evaluate the duration of in vivo persistence of adoptively transferred T cells, and the phenotype of persisting T cells. Real Time polymerase chain receptor (RT-PCR) and Flow cytometry(FCM) analysis of PB,BM and lymph node will be used to detect and quantify survival of universal dual specificity CD19 and CD20 or CD22 CAR-T cells over time.

2. SECONDARY OBJECTIVES:

   1. For patients with detectable disease, measure anti-tumor response due to universal dual specificity CD19 and CD20 or CD22 CAR-T cell infusions.
   2. Determine if cellular or humoral host immunity develops against the murine anti-CD19, and assess correlation with loss of detectable universal dual specificity CD19 and CD20 or CD22 CAR-T cells (loss of engraftment).

The CAR-T cells will be administered by i.v. injection over 20-30 minutes as a using a "split dose" approach to dosing: 10% on day 0, 30% on day 1 and 60% on day 2.

Inclusion Criteria:

1. Male or female participant
2. 12 Years to 70 Years (Child, Adult, Senior)
3. Patient with relapsed or refractory B-cell leukemia or lymphoma
4. Estimated life expectancy ≥ 12 weeks (according to investigator's judgement)
5. Eastern Cooperative Oncology Group (ECOG) performance status ≤ 1
6. Adequate organ function

Exclusion Criteria:

1. Prior malignancy, except carcinoma in situ of the skin or cervix treated with curative intent and with no evidence of active disease
2. Diagnosis of Burkitt's leukemia/lymphoma according to WHO classification or chronic myelogenous leukemia lymphoid blast crisis
3. Richter's syndrome
4. Presence of Grade II-IV (Glucksberg) or B-D (IBMTR) acute or extensive chronic GVHD at the time of screening
5. Subjects with any autoimmune disease or any immune deficiency disease or other disease in need of immunosuppressive therapy
6. Severe active infection (uncomplicated urinary tract infections, bacterial pharyngitis is allowed), Prophylactic antibiotic, antiviral and antifungal treatment is permissible
7. Active hepatitis B, active hepatitis C, or any human immunodeficiency virus (HIV) infection at the time of screening
8. Patient has an investigational medicinal product within the last 30 days prior to screening
9. Previous treatment with investigational gene or cell therapy medicine products
10. Concurrent use of systemic steroids. Recent or current use of inhaled steroids is not exclusionary
11. Pregnant or nursing women