• Ability of participant to understand this study, and participant willingness to sign a written informed consent

• Women of childbearing potential must have a negative serum pregnancy test 48 hours prior to the start of preparatory regimen

• Patients must have histologically confirmed aggressive B-Cell NHL or ALL as stated below:

• A. Patients with relapsed or refractory B-Cell ALL i. Demonstration of one or more antigens of interest (CD19, CD20, CD22) in most recent disease evaluation prior to study entry and within 30 days of study entry.

• ii. Patients with relapsed/refractory disease in blood, marrow, and extramedullary sites including CSF will be eligible when there is immunophenotypic evidence of CD19 and/or CD20 and/or CD22 expression

• iii. Primary refractory disease at study entry defined as: A morphologic complete response has never been achieved prior to study entry.

• iv. Early first relapse at study entry defined as: Disease recurrence by morphologic assessment after duration of first remission at ≤ 18 months

• v. Relapsed Refractory disease (first or later relapse) at study entry defined as: Morphologic complete response was not achieved after initiation of a second-line (or later) systemic therapy

• vi. Second or greater relapse at study entry defined as: Any disease recurrence following a second or later complete response (with treatment history including two or more lines of systemic therapy)

• vii. Additional considerations beyond above criteria:

⁃ Patients with relapsed or refractory disease after allogeneic stem cell transplantation must be \>100 days from HSCT to be eligible for study participation. Furthermore, post-HSCT immunosuppressive medications must be discontinued for at least 4 weeks prior to study entry

⁃ Prior CAR-T therapy is permissible if ≥ 3 months from therapy completion

⁃ Morphological disease in the bone marrow Note: Morphologic disease is defined as blasts being at least 5% in the bone marrow.

• B. Histologically confirmed aggressive B cell NHL, including the following types defined by WHO 2008 after 2 or more lines of prior therapy:

• i. DLBCL not otherwise specified; T cell/histiocyte rich large B cell lymphoma; DLBCL associated with chronic inflammation; Epstein-Barr virus (EBV)+ DLBCL of the elderly; Primary cutaneous DLBCL, leg type; OR ii. Primary mediastinal (thymic) large B cell lymphoma iii. Follicular lymphoma 3b and transformation of follicular lymphoma to DLBCL will also be included iv. High-grade B cell lymphoma v. Chemotherapy-refractory disease, defined as one or more of the following:

⁃ Refractory disease is defined as progressive or stable disease as the best response to the most recent prior therapy or relapse within 12 months of autologous stem cell transplantation. Two prior lines of therapy are required for LBCL eligibility. The second line therapy may be chemotherapy based, autologous stem cell transplantation, or CAR-T.

⁃ Relapsed or refractory disease after allogeneic transplant provided patient is at least 100 days from stem cell transplant at the time of enrollment and off of immunosuppressive medications for at least 4 weeks prior to enrollment

⁃ Patients must have received 2 or more lines of adequate prior therapy including at a minimum:

∙ anti-CD20 monoclonal antibody unless investigator determines that tumor is CD20- negative and

‣ an anthracycline containing chemotherapy regimen

‣ for patients with transformed FL must have received prior chemotherapy for follicular lymphoma and subsequently have chemorefractory disease after transformation to DLBCL vi. Prior CAR T therapy permissible if ≥ 3 months from the therapy vii. At least 1 measurable lesion according to the revised IWG Response Criteria for Malignant Lymphoma (Cheson 2007). Lesions that have been previously irradiated will be considered measurable only if progression has been documented following completion of radiation therapy C. Relapsed or refractory indolent non-Hodgkin lymphoma i. Histologically confirmed indolent non-Hodgkin lymphoma, including grade 1-3b follicular lymphoma or nodal or extranodal marginal zone lymphoma (both per WHO 2016 classification criteria) ii. Relapsed or refractory disease (per Lugano criteria) after two or more previous lines of therapy, iii. Previous lines of therapy to include an anti-CD20 monoclonal antibody combined with an alkylating agent iv. Prior CAR T therapy permissible if ≥ 3 months from the therapy v. At least 1 measurable lesion according to the revised IWG Response Criteria for Malignant Lymphoma (Cheson 2007). Lesions that have been previously irradiated will be considered measurable only if progression has been documented following completion of radiation therapy vi. Relapsed or refractory disease after allogeneic transplant provided patient is at least 100 days from stem cell transplant at the time of enrollment and off of immunosuppressive medications for at least 4 weeks prior to enrollment D. Relapsed or Refractory Mantle-Cell Lymphoma i. Histologically confirmed mantle-cell lymphoma with either cyclin D1 overexpression or presence of the translocation (T11:14) ii. Disease that is either relapsed or refractory to at least 2 prior lines of previous regimens for mantle-cell lymphoma iii. Previous therapy must have included anthracycline- or bendamustine-containing chemotherapy, an anti-CD20 monoclonal antibody, and BTK inhibitor therapy

• Prior CAR T therapy permissible if ≥ 3 months from the therapy

• At least 1 measurable lesion according to the revised IWG Response Criteria for Malignant Lymphoma (Cheson 2007). Lesions that have been previously irradiated will be considered measurable only if progression has been documented following completion of radiation therapy

• Relapsed or refractory disease after allogeneic transplant provided patient is ≥ 3 months from stem cell transplant at the time of enrollment and off of immunosuppressive medications for at least 4 weeks prior to enrollment

• Meet institutional criteria for leukapheresis procedure or have availability of previously- collected and stored leukapheresis product that satisfies minimum requirements

• Eastern cooperative oncology group (ECOG) performance status of 0 to 2.

• Adequate hematologic and organ function NOTE: Patients with established diagnosis of benign neutropenia are eligible to participate with ANC between 1000-1500 if in the opinion of treating physician the trial treatment does not pose excessive risk of infection to the patient.

⁃ Adults ≥ 18 years of age, with no upper limit of age

⁃ Life expectancy \>2 months

⁃ ≥ 3 months from prior CAR

⁃ Women of child-bearing potential and men with partners of child-bearing potential must agree to practice sexual abstinence or to use the forms of contraception listed in Child-Bearing Potential/Pregnancy section from the time of signing informed consent to at least 12 months following DuoCAR20.19.22-D95 infusion and until CAR positive viable T cells are no longer present by quantitative polymerase chain reaction (qPCR) on two consecutive tests. Men must agree not to donate sperm for the same time period.