Inclusion Criteria:

* Have a diagnosis of CLL requiring treatment according to the International Workshop on CLL (iwCLL) criteria (Hallek et al 2018)
* Eastern Cooperative Oncology Group (ECOG) performance score (PS) of ≤ 2
* Adequate bone marrow (BM) function independent of growth factor or transfusion support, within 2 weeks of screening, at screening as defined by the protocol unless cytopenia is clearly due to marrow involvement of CLL
* Adequate liver function unless directly attributable to the participant's CLL
* Life expectancy \> 6 months
* For women of childbearing potential: agreement to remain abstinent (refrain from heterosexual intercourse) or use contraceptive methods that result in a failure rate of \< 1% per year, and agreement to refrain from donating eggs during the treatment period and for at least 3 months after the last dose of mosunetuzumab and 3 months after the last dose of tocilizumab (if applicable)
* For men: agreement to remain abstinent (refrain from heterosexual intercourse) or use a condom, and agreement to refrain from donating sperm as defined by the protocol

Inclusion Criteria Specific to Arm B:

* Participants must have been taking a BTKi for at least 12 months, have demonstrated evidence of progressive disease while receiving the BTKi and require additional salvage therapy as assessed by their treating physician. Participants should be able to continue their previously prescribed BTKi at a stable dose throughout the study screening period and for the first two cycles of mosunetuzumab administration

Inclusion Criteria Specific to Arm C:

* Non-US participants only

Exclusion Criteria:

* Pregnant or breastfeeding, or intending to become pregnant during the study or within 3 months after the final dose of mosunetuzumab and tocilizumab or within 30 days after the final dose of venetoclax (if applicable)
* Participants who have received any of the following treatments prior to study entry: treatment with mosunetuzumab or other CD20/CD3-directed bispecific antibodies; allogenic stem cell transplant
* Participants who have received any of the following treatments, whether investigational or approved, within the respective time periods prior to initiation of study treatment: radiotherapy within 2 weeks prior to the first dose of study treatment; autologous stem cell transplant within 100 days prior to first study treatment; CAR T-cell therapy within 30 days before first study treatment; prior use of any monoclonal antibodies, radioimmunoconjugates, or antibody-drug conjugates for anti-CLL treatment within 4 weeks before first dose of study treatment; systemic immunosuppressive medications (including but not limited to cyclophosphamide, azathioprine, methotrexate, thalidomide, and anti-tumor necrosis factor agents) within 2 weeks prior to the first dose of study treatment; any other anti-cancer therapy, whether investigational or approved, including but not limited to chemotherapy within 4 weeks prior to initiation of study treatment (except for participants enrolled in Arm B, where overlapping therapy is permitted; other prior cancer immunotherapy not explicitly defined by the protocol is to be discussed with the medical monitor to determine eligibility
* Received a live, attenuated vaccine within 4 weeks before the first dose of study treatment, or in whom it is anticipated that such a vaccine will be required during the study period or within 5 months after the final dose of study treatment
* Transformation of CLL to aggressive non-Hodgkin's lymphoma (NHL)
* History of severe allergic or anaphylactic reactions to humanized or murine monoclonal antibody therapy (or recombinant antibody-related fusion proteins)
* Contraindication to tocilizumab
* History of prior malignancy except for conditions defined by the protocol
* Participants with infections requiring intravenous (IV) treatment with antibiotics or hospitalization within the last 4 weeks prior to enrollment or known active bacterial, viral (including SARS-CoV-2), fungal, mycobacterial, parasitic, or other infection (excluding fungal infections of nail beds) at study enrollment
* Evidence of any significant concomitant disease that could affect compliance with the protocol or interpretation of results
* Recent major surgery within 4 weeks prior to first study treatment administration, with the exception of protocol-mandated procedures (e.g., tumor biopsies and bone marrow biopsies)
* Positive SARS-CoV-2 test within 7 days prior to enrollment

Exclusion Criteria Specific to Arm C:

* Have received venetoclax therapy within 12 months prior to first study treatment administration
* Participants with known infection with HIV or human T-cell leukemia virus 1 (HTLV1)
* HIV testing will be performed in countries where mandatory testing by health authorities is required
* HTLV testing is required in participants from endemic countries
* Participants with uncontrolled autoimmune hemolytic anemia or immune thrombocytopenia
* Participants who have received the following: strong and moderate CYP3A inhibitors within 7 days prior to the initiation of study treatment; strong and moderate CYP3A inducers within 7 days prior to the initiation of study treatment; steroid therapy for anti-neoplastic intent with the exception of inhaled steroids for asthma, topical steroids, or replacement/stress corticosteroids within 7 days prior to the first dose of study drug administration
* Have consumed grapefruit, grapefruit products, Seville oranges (including marmalade containing Seville oranges), or star fruit within 3 days prior to the first dose of study drug and throughout venetoclax administration
* Inability to swallow a large number of tablets
* Malabsorption syndrome or other condition that precludes enteral route of administration
* Known allergy to both xanthine oxidase inhibitors and rasburicase