• Cohort A: Previously untreated MZL. Prior therapy with H. Pylori antibiotic therapy or hepatitis C antiviral therapy are allowed on Cohort A.

• Cohort B: Previously untreated FL

• Pathological confirmation of lymphoma: availability of archival tissue confirming diagnosis of MZL (cohort A) or FL (cohort B). Availability of formalin-fixed, paraffin-embedded (FFPE) archival tumor specimens from within past 18 months from screening and pathological diagnosis confirmed by a pathologist at the participating site or willingness of the participant to undergo a fresh tumor biopsy if adequate archival tissue not available is required. This includes:

• MZL (Cohort A):

‣ Nodal MZL requiring systemic therapy

⁃ Splenic MZL requiring systemic therapy

⁃ Extra-nodal marginal zone lymphoma:

• Non-gastric/non-cutaneous MZL requiring systemic therapy.

∙ Cutaneous MZL will be eligible only if they have pathologically confirmed extra-cutaneous disease.

∙ Gastric MZL only if advanced stage disease requiring systemic therapy (e.g., stage IIE, II2, IV- supradiaphragmatic nodal or disseminated extranodal disease such as bone marrow or additional extra nodal sites.

• FL (Cohort B):

• a. Pathological grade 1, 2, or 3a based on the World Health Organization (WHO 2008) classification of tumors of hematopoietic and lymphoid tissue.

• 1\. Please note, grade 3B are excluded.

• All participants must have disease requiring systemic therapy rather than local radiation (ie, stage II only if not eligible for radiation therapy or with stage III/IV).

• All participants should have measurable disease. Measurable disease is defined as a lymph node or tumor mass that is ≥ 1.5 cm in at least one dimension by CT or the CT portion of the PET/CT.

• Documentation of CD20+ status.

• All participants must have an indication for therapy such as: symptoms attributable to lymphoma, threatened end-organ function, or cytopenia secondary to lymphoma.

• All participants must be 18 years of age or older.

• All participants must have an Eastern Cooperative Oncology Group (ECOG) Performance Status of 0-2.

• All participants must be able to swallow whole pills.

• All participants must have the ability and willingness to comply with the requirements of the study protocol.

• All female participants who are of non-reproductive potential (i.e., post-menopausal by history - no menses for ≥1 year; OR history of hysterectomy; OR history of bilateral tubal ligation; OR history of bilateral oophorectomy).

• All female participants of childbearing potential must have a negative serum pregnancy test upon study entry.

• All male and female participants of reproductive potential who agree to use both a highly effective method of birth control (e.g., implants, injectables, combined oral contraceptives, some intrauterine devices \[IUDs\], complete abstinence, or sterilized partner) and a barrier method (e.g., condoms, vaginal ring, sponge, etc) during the period of therapy. Female participants of reproductive potential who are not surgically sterile must practice adequate birth control for a minimum of 30 days after last dose of zanubrutinib or 12 months after last dose of rituximab, whichever is longer. Male participants are eligible if abstinent, vasectomized, or if they agree to the use of barrier contraception in combination with other methods described above during the study treatment period and for ≥ 30 days after the last dose of zanubrutinib, or 12 months after the last dose of rituximab, whichever is longer.

• All participants must have adequate organ function.