• Subjects aged ≥ 18 years.

• ECOG Performance Status ≤ 2.

• Histologically confirmed marginal zone lymphoma, including splenic, nodal, and extranodal sub-types per the enrolling institution.

• Subjects must have an indication for treatment.

• No prior systemic therapy for MZL except for the following:

‣ Prior antibiotic therapy for H. pylori, C. psittaci, and B. burgdorferi

⁃ Prior antiviral therapy for HCV

∙ --Note: Subjects are eligible if they had a prior splenectomy or other local surgical treatment or local radiation therapy without systemic therapy and now require their first ever systemic therapy. In the event of the receipt of radiation therapy, the minimum washout period is 14 days

• Subjects with gastric MALT lymphoma must be H. pylori negative or have failed a trial of H. pylori eradication

• Subjects with localized MALT lymphoma must be ineligible for, have refused or failed radiation therapy (washout period of 14 days)

• Adequate organ function as defined as:

‣ Hematologic:

• Absolute neutrophil count ≥ 750 cells/mm3 (≥ 0.75 x 10\^9/L) independent of G-CSF support, unless there is documented bone marrow involvement or splenomegaly with ensuing cytopenia in which case ANC of 500 cells/mm3 (0.5 x 10\^9/L) is permissible. Also, there should be no evidence of myelodysplasia or hypoplastic bone marrow

∙ Platelet count ≥ 50,000/mm3 (≥50 x 10\^9/L) independent of transfusion support unless there is documented bone marrow involvement in which case platelet count of ≥30,000 cells/mm3 (≥30 x 10\^9/L) is permissible. Subjects must be responsive to transfusion support if given for thrombocytopenia and subjects refractory to transfusion support are not eligible. Also, there should be no evidence of myelodysplasia or hypoplastic bone marrow.

‣ Note: The platelet count threshold in the current study (≥50,000 cells/mm\^3 or ≥50 x 10\^9/L) is lower than normal threshold (≥75,000 cells/mm\^3 or ≥75 x 10\^9/L) as the majority of MZL subjects have lower than normal platelets due to splenomegaly and or autoimmune phenomena (which are related to the underlying lymphoma) and hence the lower than normal platelet count threshold for study entry

∙ Hemoglobin ≥ 8 g/dL independent of transfusion support unless there is documented bone marrow involvement or splenomegaly with ensuing cytopenia in which case hemoglobin of ≥7 g/dL (≥70 g/L) is permissible. Subjects must be responsive to transfusion support if given for anemia and subjects refractory to transfusion support are not eligible. Also, there should be no evidence of myelodysplasia or hypoplastic bone marrow.

∙ Hepatic:

⁃ ---Total Bilirubin ≤ 1.5x institutional upper limit of normal (ULN)

⁃ ----Subjects with liver involvement will be allowed to enroll with total bilirubin ≤3 x ULN

∙ AST(SGOT)/ALT(SGPT) ≤ 3 × institutional ULN ----Subjects with liver metastases will be allowed to enroll with AST and ALT levels ≤ 5 x ULN.

‣ Renal: Estimated creatinine clearance ≥ 30 mL/min by Cockcroft-Gault formula:

∙ aPPT or PT or INR ≤ 1.5 X ULN

• Life expectancy of \>3 months, in the opinion of the investigator

• For female subjects: Negative pregnancy test or evidence of post-menopausal status. The post-menopausal status will be defined as having been amenorrheic for 12 months without an alternative medical cause. The following age-specific requirements apply:

‣ Women \< 50 years of age:

∙ --Amenorrheic for ≥ 12 months following cessation of exogenous hormonal treatments; and

∙ Luteinizing hormone and follicle-stimulating hormone levels in the post-menopausal range for the institution; or

∙ Underwent surgical sterilization (bilateral oophorectomy or hysterectomy).

⁃ Women ≥ 50 years of age:

• Amenorrheic for 12 months or more following cessation of all exogenous hormonal treatments; or

∙ Had radiation-induced menopause with last menses \>1 year ago; or

∙ Had chemotherapy-induced menopause with last menses \>1 year ago; or

∙ Underwent surgical sterilization (bilateral oophorectomy, bilateral salpingectomy, or hysterectomy).

• Female subjects of childbearing potential and male subjects with a sexual partner of childbearing potential must agree to use a highly effective method of contraception as described in Section 5.3.1.

• Subjects may not plan to become pregnant or breastfeed within 1 month of the last dose of pirtobrutinib or 12 months following the last rituximab infusion

• Ability to swallow oral tablets.

• Subjects or their legal representatives must be able to read, understand, and provide informed consent to participate in the trial.