* INCLUSION CRITERIA:
Malignancy criteria
* CD19 and or CD20 expression is required. For all lymphoma types, eligibility criteria are met if there is uniform expression of both CD19 and CD20. Eligibility criteria are also met with expression of either CD19 or CD20. CD19 and/or CD20 expression must be "uniform". "Uniform" CD19 or CD20 expression is defined by CD19 and/or CD20 antigen expression on lymphoma cells with no obvious lymphoma population lacking antigen expression. Antigen expression can be assessed by either immunohistochemistry or flow cytometry.
* Pathology confirmed B-cell malignancy. 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.
* At least 14 days must elapse between the time of any prior systemic treatment (including corticosteroids) and protocol-required leukapheresis or start of protocol conditioning chemotherapy.
* At least sixty days must elapse from therapy with antibodies targeting CD19 or CD20 and CAR T-cell infusion.
* At least 180 days must elapse after any prior CAR T-cell therapy, but otherwise, prior CAR T-cell therapy is allowed.
* Participants with DLBCL (including all subtypes such as primary mediastinal B-cell lymphoma and high-grade B-cell lymphoma with MYC and BCL2 and/or BCL6 rearrangement) must have received at least two prior regimens at least one of which must have contained doxorubicin and an anti-CD20 monoclonal antibody.
* Follicular lymphoma participants must have received at least 2 prior regimens including at least 1 regimen with chemotherapy and 1 regimen with an anti-CD20 monoclonal antibody.
* Burkitt lymphoma participants must have had at least 1 prior cytotoxic chemotherapy- containing regimen that also contained an anti-CD20 monoclonal antibody.
* All participants with CLL or small lymphocytic lymphoma must have had at least 1 prior line of treatment, and these participants must have had progressive CLL/SLL after exposure to ibrutinib or another Bruton tyrosine kinase inhibitor and venetoclax.
* T-cell rich B-cell lymphoma is eligible if previously treated with at least 2 lines of therapy.
* Waldenstrom s macroglobulinemia/lymphoplasmacytic lymphoma patients are eligible if previously treated with at least 2 regimens including exposure to a monoclonal antibody, a bruton tyrosine kinase inhibitor, and cytotoxic chemotherapy.
* Participants with mantle cell lymphoma are eligible after receiving a Bruton tyrosine kinase (BTK) inhibitor, an anti-CD20 monoclonal antibody, and at least one of the following chemotherapy drugs: cytarabine, bendamustine, or an anthracycline.
* Other B-cell lymphoma types, including B-cell lymphoma unclassifiable with features intermediate between DLBCL and Hodgkin lymphoma (Gray zone), that are not specifically mentioned above are allowed if the participant has received at least 2 lines of therapy at least 1 of which must have contained cytotoxic chemotherapy.
* Primary central nervous system lymphoma patients are not eligible.
All participants 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 unless bone marrow or blood involvement with malignancy is detected. 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. Exceptions to this rule are malignancies that do not consistently display increased metabolic activity on PET scans such as CLL/Small lymphocytic lymphoma.
* For CLL and lymphoma with only bone marrow and/or blood involvement no mass is necessary, but if a mass is not present, bone marrow and/or blood malignancy must be detectable by flow cytometry. Note that leukemia cells must make up 1% or less of peripheral blood lymphocytes within 2 weeks of the time of protocol enrollment in CLL participants for CLL participants to be eligible.
Other inclusion criteria:
* Greater than or equal to 18 years of age and less than or equal to 75 years of age.
* Clinical performance status of ECOG 0-1.
* 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
* Serum ALT and AST less or equal to 3 times the upper limit of the institutional normal unless liver involvement by malignancy is demonstrated. If liver involvement with malignancy is detected, ALT and AST must be less than or equal to 5 times the upper limit of normal.
* Serum creatinine less than or equal to 1.5 mg/dl
* Total bilirubin less than or equal to 2.0 mg/dl
* Room air oxygen saturation of 92% or greater
* Participants 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 months after receiving the protocol treatment.
* A participant with a negative blood PCR test for hepatitis B DNA test can be enrolled. If hepatitis B DNA (PCR) testing is not available, participants with a negative hepatitis B surface antigen and negative hepatitis B core antibody can be enrolled.
* Participants 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 ina timely manner, participants who are Hepatitis C antibody-negative can be enrolled.
* Cardiac ejection fraction of 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.
* Participants must be able to understand and be willing to sign a written informed consent.
* Participants who have either been previously treated with genetically-modified T- cells including CAR T -cells of any specificity are potentially eligible if at least 180 days have elapsed since the prior infusion of genetically-modified T- cells. An exception to this is patients previously treated on this protocol can be re-treated on this protocol 8 weeks or more after the first treatment on this protocol.
EXCLUSION CRITERIA:
* Participants that require urgent therapy due to tumor mass effects or spinal cord compression.
* Participants must not have received any anti-CD20 or anti-CD19 antibody products in the past 60 days prior to CAR T-cell infusion.
* Participants that have active hemolytic anemia.
* HIV-positive patients.
* Participants 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.
* Currently pregnant (confirmed with beta-HCG serum or urine pregnancy test performed at screening) or breastfeeding.
* Active uncontrolled systemic infections (defined as infections causing fevers within 48 hours of the date of planned protocol chemotherapy start and infections requiring intravenous antibiotics when intravenous antibiotics have been administered for less than 72 hours at the time of protocol chemotherapy start).
* 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.
* Significant neurologic disorders that are not completely and permanently resolved and not requiring current treatment.
* Any form of primary immunodeficiency (such as Severe Combined Immunodeficiency Disease).
* Prior allogeneic stem cell transplant
* Systemic 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.
* Participants on systemic anticoagulant therapy except aspirin.
* History of severe immediate hypersensitivity reaction to any of the agents used in this study.
* Active central nervous system/brain metastases or cerebrospinal fluid malignancy.
* Checkpoint inhibitor drugs such as pembrolizumab or nivolumab or other antibodies targeting PD-1 or PDL-1 within 180 days of protocol enrollment. This is because of possible effects checkpoint inhibitor therapy could have on the patient s T- cells.
