Durvalumab and Tremelimumab in Treating Participants With Recurrent or Refractory Ovarian, Primary Peritoneal, or Fallopian Tube Cancer

PRIMARY OBJECTIVES: I. To determine the median immune-related progression-free survival (irPFS) in the experimental arms. SECONDARY OBJECTIVES: I. To determine the rate of grade III or higher treatment related toxicity in each experimental arm. II. To describe the immunological and gene expression changes induced by tremelimumab and the combination of tremelimumab and durvalumab in epithelial ovarian cancer (EOC) tumor tissues and blood. III. To determine the overall survival (OS), objective response rate (ORR) IV. To determine the proportion of patients that discontinue treatment due to side effects. EXPLORATORY OBJECTIVES: I. To determine second progression-free survival (PFS) (PFS2) following initial progression in each arm. II. To determine the response rate to durvalumab (MEDI4736) following treatment with tremelimumab (in the sequential arm). III. To evaluate the patient reported symptom burden in each experimental arm. To better assess the relationship between somatic tumor mutations in the PPP2R1A gene and response and survival following combination therapy with tremelimumab and durvalumab in two molecularly defined expansion cohorts: (a) subjects with ovarian clear cell carcinoma and (b) subjects with uterine serous carcinoma. (This objective will represent an expansion cohort in Arm 2 and will be independent of enrollment to the main study) OUTLINE: Participants are randomized to 1 of 2 arms. ARM I (SEQUENTIAL): Patients receive tremelimumab intravenously (IV) over 60 minutes on day 1. Treatment repeats every 4 weeks for 4 cycles in the absence of disease progression or unacceptable toxicity. Upon progression, patients then receive durvalumab IV over 60 minutes on day 1. Treatment repeats every 4 weeks for up to 9 cycles in the absence of disease progression or unacceptable toxicity. This arm is closed to enrollment. ARM II (COMBINATION): Patients receive tremelimumab IV and durvalumab IV over 60 minutes (for each drug) on day 1. Treatment repeats every 4 weeks for 4 cycles in the absence of disease progression or unacceptable toxicity. Patients then receive durvalumab IV over 60 minutes on day 1. Treatment repeats every 4 weeks for up to 9 cycles in the absence of disease progression or unacceptable toxicity. After completion of study treatment, patients are followed up at 30 days and then every 2 months thereafter