Atezolizumab and Bevacizumab in Treating Patients With Recurrent, Persistent, or Metastatic Cervical Cancer

PRIMARY OBJECTIVES: I. To assess the anti-tumor activity (proportion of patients with objective response by Response Evaluation Criteria in Solid Tumors version 1.1 \[RECIST 1.1\] criteria) of atezolizumab and bevacizumab in patients with recurrent, persistent or metastatic cervical cancer. SECONDARY OBJECTIVES: I. To estimate the duration of progression free survival (PFS) and overall survival (OS). II. To assess safety by Common Terminology Criteria for Adverse Events version 5.0 (CTCAE v.5.0). III. To describe the efficacy of the combination of atezolizumab and bevacizumab as measured by objective response, by PD-L1 expression on tumor and immune cells measured by semi-quantitative immunohistochemistry (IHC). IV. To describe the efficacy of the combination of atezolizumab and bevacizumab as measured by objective response, by intratumoral and peripheral T-cell receptor (TCR) clonality and tumor infiltrating lymphocyte proportion. OUTLINE: Patients receive atezolizumab intravenously (IV) over 30-60 minutes and bevacizumab IV over 30-90 minutes on day 1. Courses repeat every 21 days in the absence of disease progression or unacceptable toxicity. After completion of study treatment, patients are followed up every 3 months for 2 years.