Inclusion Criteria:

          -  Participants receiving a histologic diagnosis of epithelial ovarian cancer (EOC),
             peritoneal primary carcinoma, or fallopian tube cancer

          -  Eastern Cooperative Oncology Group (ECOG) performance status of 0, 1, or 2

          -  Life expectancy greater than (>) 12 weeks

          -  For participants who receive therapeutic anticoagulation: stable anticoagulant regimen

          -  Availability of a representative formalin-fixed, paraffin-embedded (FFPE) tumor
             specimen in paraffin blocks (preferred) or at least 20 unstained slides (for detailed
             tissue requirements at screening)

        Exclusion Criteria:

          -  Received a current diagnosis of borderline epithelial ovarian tumor (formerly tumors
             of low malignant potential)

          -  Have recurrent invasive epithelial ovarian, fallopian tube, or primary peritoneal
             cancer that was treated only with surgery (example [e.g.], participants with Stage IA
             or Stage IB epithelial ovarian or fallopian tube cancers)

          -  Have non-epithelial ovarian tumors (e.g., germ cell tumors, sex cord stromal tumors)

          -  Received prior radiotherapy to any portion of the abdominal cavity or pelvis

          -  Received prior chemotherapy for any abdominal or pelvic tumor that include neoadjuvant
             chemotherapy (NACT) for ovarian, primary peritoneal or fallopian tube cancer

          -  Received any biological and/or targeted therapy (including but not limited to
             vaccines, antibodies, tyrosine kinase inhibitors) or hormonal therapy for management
             and/or treatment of epithelial ovarian or peritoneal primary cancer

          -  Have synchronous primary endometrial cancer

          -  Have a prior history of primary endometrial cancer, except: Stage IA cancer;
             superficial myometrial invasion, without lymphovascular invasion; grade less than (<)
             3 or poorly differentiated subtypes, and this includes papillary serous, clear cell or
             other International Federation of Gynecological Oncologists (FIGO) Grade 3 lesions

          -  With the exception of non-melanoma skin cancer and other specific malignancies as
             noted above, other invasive malignancies with any evidence of other cancers present
             within the last 5 years or previous cancer treatment that contraindicates this
             protocol therapy

          -  Have a known hypersensitivity or allergy to biopharmaceutical agents produced in
             Chinese hamster ovary cells or any component of the atezolizumab and/or bevacizumab
             formulations

          -  Undergo major surgical procedure within 28 days prior to first bevacizumab dose, or
             anticipation of the need for a major surgical procedure during the course of the study
             except participants who receive NACT and will need interval surgery. This may include
             but is not limited to laparotomy.

          -  Have prior allogeneic bone marrow transplantation or solid organ transplant

          -  Have any other diseases, metabolic dysfunction, physical examination finding, or
             clinical laboratory finding giving reasonable suspicion of a disease or condition that
             contraindicates the use of an investigational drug or that may affect the
             interpretation of the results

          -  Have any approved or investigational anti-cancer therapy, including chemotherapy or
             hormonal therapy, with exceptions: Hormone-replacement therapy or oral contraceptives

          -  Are administered treatment with any other investigational agent or participation in
             another clinical study with anti-cancer therapeutic intent

          -  Have core biopsy or other minor surgical procedures within 7 days prior to the first
             dose of bevacizumab

          -  Have known sensitivity to any component of bevacizumab

          -  Have known sensitivity to any component of paclitaxel

          -  Current treatment with anti-viral therapy for hepatitis B virus (HBV)

          -  History of leptomeningeal disease