Drug combo aims to boost remission in rare lymphoma

Copied word for word from the study's registry entry, so the wording is the study team's rather than ours.

Inclusion Criteria: * Age greater than or equal to 18 years * Histologically confirmed Marginal Zone Lymphoma * Patients must have measurable disease as defined by at least one lymph node ≥1.5 cm or spleen \>13 cm. °Patients with intestinal MALT lymphoma must have disease that is detectable by EGD or colonoscopy with biopsy * Patients with gastric MALT lymphoma must be h. pylori negative °Patients who are h. pylori positive are allowed if they have failed a trial of h.pylori eradication * Patients with gastric MALT lymphoma who are h. pylori negative or who have relapsed/refractory disease after h. pylori eradication must be ineligible for, have refused or failed gastric radiation therapy * ECOG performance status ≤ 1 * Life expectancy of greater than 2 years * Patients must have normal organ function as defined below: * Platelet count ≥ 50,000 cells/mm\^3 * Hemoglobin ≥ 8.0 g/dL * Absolute neutrophil count ≥ 1000 cells/mcL. If there is documented bone marrow involvement, ANC must be \>/= 500 cells/mcL * Total bilirubin \< 1.5 x upper normal institutional limits. In patients with Gilbert's disease or documented liver involvement, total bilirubin up to 3x ULN will be allowed * AST(SGOT)/ALT(SGPT) \<3 x institutional upper limit of normal unless elevation is caused by liver involvement with MZL * AST(SGOT)/ALT(SGPT) \<3 x institutional upper limit of normal unless elevation is caused by liver involvement with MZL °OR Creatinine clearance \>60 mL/min for patients with creatinine levels above institutional normal (by Cockcroft-Gault estimate or 12-24h creatinine clearance measurements). * Ability to understand and the willingness to sign a written informed consent document. * Able to swallow pills * HIV-positive patients on combination antiretroviral therapy are eligible if their HIV is under adequate control with an antiretroviral regimen that has been stable for \> 4 weeks, as long as the CD4 count is \> 300. Appropriate studies will be undertaken in patients receiving combination antiretroviral therapy when indicated. * Patients with Hepatitis B surface antibody serum positivity due to prior immunization, as well as those with Hepatitis B core antibody positivity with negative PCR on antiviral therapy will be eligible. Exclusion Criteria: * Patients who have had prior systemic therapy, including rituximab * Patients who have had prior radiation therapy, with the following exception: °Palliative radiotherapy RT is allowed but must be completed at least 1 week prior to treatment on this study, and prior baseline imaging studies or biopsies. Patients must meet criteria for measurable/assessable disease as outlined above after completion of RT * Prior treatment with ibrutinib or other BTK inhibitor * Patients with h. pylori-associated gastric MALT or stage I/II MZL will be excluded unless they are deemed to be unfit for radiation therapy with curative intent. * Patients with uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, or psychiatric illness/social situations that would limit compliance with study requirements °Patients with Hep B core ab positivity are allowed provided Hep B PCR is undetectable * Lactating or pregnant women * Participants unwilling to adhere to institutional guidelines for highly effective contraception for 12 months after the last dose of rituximab * Patients who received moderate or strong CYP3A inhibitors (such as fluconazole, ketoconazole, and clarithromycin) within 7 days prior to the first dose of venetoclax. * Patients who received moderate or strong CYP3A inducers (such as rifampin, carbamazepine, phenytoin, St. John's Wort) within 7 days prior to the first dose of venetoclax.