-  INCLUSION CRITERIA:

        MALIGNANCY CRITERIA:

        Note: As of approval of Amendment A, no patients with Hodgkin lymphoma can be enrolled
        until at least 6 patients with B-cell malignancies are treated without incidence of
        Guillain-Bare syndrome

          -  Patients must have any B-cell lymphoma, or CLL/SLL, Gray-zone lymphoma, nodular
             lymphocyte-predominant Hodgkin lymphoma, or classical Hodgkin lymphoma with any CD19
             or CD20 expression on Reed-Sternberg cells. Lower grade lymphomas or CLL transformed
             to DLBCL are potentially eligible as is primary mediastinal B-cell lymphoma and all
             other subtypes of DLBCL. Burkitt and mantle cell lymphoma are potentially eligible.

          -  For classical Hodgkin lymphoma only, a biopsy from any time from any institution that
             shows any CD19 or CD20 expression on Reed-Sternberg cells is adequate for eligibility.
             CD19 or CD20 expression on the Reed-Sternberg cells that is weak or only present on
             some Reed-Sternberg cells by immunohistochemistry is compatible with protocol
             eligibility.

          -  For all lymphoma types except for classical Hodgkin lymphoma, either CD19 or CD20
             expression must be uniform . Uniform CD19 or CD20 expression is defined as no obvious
             lymphoma population lacking antigen expression is present. Antigen expression can be
             assessed by either immunohistochemistry or flow cytometry.

          -  Only when insufficient biopsy material is available to allow CD19 and CD20 expression
             assessment at the NIH, CD19 and/or CD20 staining performed at another institution can
             be used

          -  DLBCL patients must have received at least two prior chemotherapy-containing regimens
             at least one of which must have contained doxorubicin and a monoclonal antibody.
             Follicular lymphoma patients must have received at least 2 prior regimens including at
             least 1 regimen with chemotherapy. All other B-cell lymphoma and leukemia patients
             must have had at least 1 prior chemotherapy-containing regimen. All patients with CLL
             or small lymphocytic lymphoma must have had prior treatment with ibrutinib or another
             signal transduction inhibitor.and venetoclax.

          -  Hodgkin lymphoma patients must have:

               -  had at least 3 prior lines of therapy.

               -  had at least 1 prior cytotoxic chemotherapy-containing regimen.

               -  had prior exposure to brentuximab vedotin.

               -  had undergone autologous stem cell transplant or been transplant ineligible or
                  refused autologous transplantation

          -  Eligibility will be expanded to include CD19 and CD20-negative classical Hodgkin
             lymphoma if any 2 patients with classical Hodgkin lymphoma and CD19/CD20 expression on
             RS cells have durations of response 6 months or greater (responses could be PRs or
             CRs) or a CR of 3 months or greater.

          -  All patients must have measurable malignancy as defined by at least one of the
             criteria below.

          -  Lymphoma or leukemia masses that are measurable (minimum 1.5 cm in largest diameter)
             by CT scan is required for all diagnoses except CLL. All masses must be less than or
             equal to 10.0 cm in the largest diameter.

          -  For a lymphoma mass to count as measurable malignancy, it must have abnormally
             increased metabolic activity when assessed by positron emission tomography (PET) scan.
             CLL masses do not need to have increased activity on PET scan.

          -  For CLL and lymphoma with only bone marrow involvement no mass is necessary, but if a
             mass is not present, bone marrow malignancy must be detectable by flow cytometry in
             lymphoma and CLL. Note that leukemia cells must make up 1% or less of peripheral blood
             lymphocytes in CLL patients for these patients to be eligible.

        OTHER INCLUSION CRITERIA:

          -  Greater than or equal to 18 years of age.

          -  Able to understand and sign the Informed Consent Document.

          -  Clinical performance status of ECOG 0-1

          -  Room air oxygen saturation of 92% or greater

          -  Patients of both sexes must be willing to practice birth control from the time of
             enrollment on this study and for four months after receiving the protocol treatment.

          -  A patient with a negative blood PCR test for hepatitis B DNA test can be enrolled. If
             hepatitis B DNA (PCR) testing is not available, patients with a negative hepatitis B
             surface antigen and negative hepatitis B core antibody can be enrolled.

          -  Patients must be tested for the presence of Hepatitis C antigen by PCR and be HCV RNA
             negative in order to be eligible. Only if Hepatitis C PCR testing is not available in
             a timely manner, patients who are Hepatitis C antibody-negative can be enrolled.

          -  Absolute neutrophil count greater than or equal to 1000/mm^3 without the support of
             filgrastim or other growth factors.

          -  Platelet count greater than or equal to 50,000/mm^3 without transfusion support

          -  Hemoglobin greater than 8.0 g/dl.

          -  For CLL only, less than or equal to 1% malignant cells in the peripheral blood
             lymphocytes must be documented by flow cytometry of blood within 2 weeks of protocol
             enrollment.

          -  Serum ALT and AST less or equal to 3 times the upper limit of the institutional normal
             unless liver involvement by malignancy is demonstrated.

          -  Serum creatinine less than or equal to 1.5 mg/dl.

          -  Total bilirubin less than or equal to 2.0 mg/dl.

          -  Normal cardiac ejection fraction (greater than or equal to 50% by echocardiography)
             and no evidence of hemodynamically significant pericardial effusion as determined by
             an echocardiogram within 4 weeks of treatment start.

          -  Patients must not take corticosteroids including prednisone, dexamethasone or any
             other corticosteroid for 14 days before apheresis and CAR T-cell infusion. Patients
             must also not take corticosteroids at doses higher than 5 mg/day of prednisone or
             equivalent at any time after the CAR T cell infusion.

          -  Patients must be able to understand and be willing to sign a written informed consent.

          -  Patients who have either been previously treated on protocols of genetically-modified
             T cells on a clinical trial at the NCI or received T cells modified with the MSGV or
             MSGV1 gamma-retroviral vectors at any institution are potentially eligible under these
             conditions:

               -  At least 3 months have elapsed since the last genetically-modified T-cell therapy
                  that the patient received, and there is no evidence of replication-competent
                  retroviruses (evidence must be provided from prior protocol Principal
                  Investigator), and persisting genetically-modified T cells are either not
                  detectable in the patient s blood or detectable at levels less than or equal to
                  0.2% of blood T cells as measured by flow cytometry using the Kip-1 antibody in
                  the flow cytometry lab of the NCI Laboratory of Pathology (Maryalice
                  Stetler-Stevenson s lab).

        EXCLUSION CRITERIA:

          -  Patients that require urgent therapy due to tumor mass effects or spinal cord
             compression.

          -  Patients must not have received any anti-CD20 or anti-CD19 antibody products in the
             past 30 days prior to CAR T-cell infusion.

          -  Any history of receiving PD-1 or PD-L1inhibitors

          -  Patients that have active hemolytic anemia.

          -  HIV-positive patients are excluded because HIV causes complicated immune deficiency
             and study treatment can pose more risks for these patients.

          -  Patients with second malignancies in addition to their B-cell malignancy are not
             eligible if the second malignancy has required treatment (including maintenance
             therapy) within the past 3 years or is not in complete remission. There are two
             exceptions to this criterion: successfully treated non-metastatic basal cell or
             squamous cell skin carcinoma.

          -  Women of child-bearing potential who are pregnant or breastfeeding because of the
             potentially dangerous effects of the preparative chemotherapy on the fetus or infant.

        Pregnant women are excluded from this study because study therapy can cause fetal harm.
        Because there is potential risk for adverse events in nursing infants secondary to
        treatment of the mother with study therapy, breastfeeding should be discontinued if the
        mother is treated with study drugs.

          -  Active uncontrolled systemic infections (defined as infections causing fevers and
             infections requiring intravenous antibiotics when intravenous antibiotics have been
             administered for less than 72 hours), active coagulation disorders or other major
             uncontrolled medical illnesses of the cardiovascular, respiratory, endocrine, renal,
             gastrointestinal, genitourinary or immune system, history of myocardial infarction,
             history of ventricular tachycardia or ventricular fibrillation, active cardiac
             arrhythmias (active atrial fibrillation is not allowed, resolved atrial fibrillation
             not requiring current treatment is allowed (anticoagulants count as current treatment)
             ), active obstructive or restrictive pulmonary disease, active autoimmune diseases
             such as rheumatoid arthritis.

          -  Hospitalization within the 7 days prior to enrollment.

          -  Any form of primary immunodeficiency (such as Severe Combined Immunodeficiency
             Disease).

          -  Prior allogeneic stem cell transplant

          -  Systemic lymphoma treatment of any type and corticosteroid steroid therapy of any dose
             greater than 5 mg/day or more of prednisone or equivalent is not allowed within 14
             days prior to the required leukapheresis, or the initiation of the conditioning
             chemotherapy regimen. Corticosteroid creams, ointments, and eye drops are allowed.

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

          -  Patients on systemic anticoagulant therapy except aspirin.

          -  Active central nervous system metastases or cerebrospinal fluid malignancy. Patients
             with known brain metastases will be excluded from this clinical trial because of their
             poor prognosis and because they often develop progressive neurologic dysfunction that
             would confound the evaluation of neurologic and other adverse events.

          -  Any current neurologic disorders except migraine headaches.