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

Inclusion Criteria: 1. Written informed consent and any locally-required authorization (e.g., HIPAA in the USA, EU Data Privacy Directive in the EU) obtained from the subject prior to performing any protocol-related procedures, including screening evaluations. 2. Age \>/= 18 years at time of study entry. 3. Eastern Cooperative Oncology Group (ECOG) performance status of 0 or 1. 4. Adequate normal organ and marrow function as defined by: Hemoglobin \>/= 9.0 g/dL (transfusion is allowed to correct anemia); Absolute neutrophil count (ANC) \>/= 1.5 x 10\^9/L (\> 1500 per mm\^3); Platelet count \>/= 100 x 10\^9/L (\>100,000 per mm\^3); Serum bilirubin \</= 1.5 x institutional upper limit of normal (ULN) unless diagnosed with Gilbert's syndrome; AST and ALT \</= 2.5 x ULN unless liver metastases are present, in which case it must be \</= 5x ULN; Serum creatinine CL \>40 mL/min by the Cockcroft-Gault formula ( Cockcroft and Gault 1976) or by 24-hour urine collection for determination of creatinine clearance: Creatinine CL (mL/min) = \[Weight (kg) x (140 - Age) x 0.85\]/\[72 x serum creatinine (mg/dL)\]. 5. Subjects must either be of non-reproductive potential (i.e., post-menopausal by history: \>/= 60 years old and no menses for \>/= 1 year without an alternative medical cause; OR history of hysterectomy, OR history of bilateral tubal ligation, OR history of bilateral oophorectomy) or must have a negative serum pregnancy test upon study entry. 6. Subject is willing and able to comply with the protocol for the duration of the study including undergoing treatment and scheduled visits and examinations including biopsies and follow up. 7. Histology (reviewed at MDACC) showing recurrent high grade epithelial ovarian, peritoneal, or fallopian tube cancer of clear cell histology or mixed carcinoma with a clear cell component. In addition, subjects for the molecularly defined expansion cohort will also include patients with recurrent uterine serous carcinoma (who have received at least one prior line of platinum-based chemotherapy), and must have somatic mutations testing (performed by MD Anderson or a CLIA certified test offered by a commercial vendor such as Foundation One or Caris) that demonstrates a nonsynonymous PPP2R1A mutation (other protein phosphatase 2A or related pathway mutations/alterations may be eligible if approved by the PI). Synonymous mutations and variants of uncertain significance do not qualify. 8. Platinum resistant or refractory disease as defined by progression of disease on a platinum-containing regimen or recurrence of disease within 180 days of previous platinum treatment (not applicable to the uterine serous carcinoma molecularly defined expansion cohort, however, subjects with uterine serous must have received at least one prior line of platinum-based therapy). 9. Have measurable disease based on modified RECIST 1.1. For the purposes of this study measurable disease is defined at least one "target" lesion that can be accurately measured in at least one dimension (longest dimension to be recorded). Each target lesion must be \>20 mm when measured by conventional techniques, including palpation, plain x-ray, CT, and MRI, or \>10 mm when measured by spiral CT. The target lesion must be distinct from other tumor areas selected for pre-treatment biopsies. Pre-treatment imaging must be performed within 4 weeks of starting therapy. Exclusion Criteria: 1. Involvement in the planning and/or conduct of the study (applies to both AstraZeneca staff and/or staff at the study site). 2. Previous enrollment or randomization in the present study. 3. Participation in another clinical study with an investigational product administered during the last 28 days. 4. Any previous treatment with adoptive T cells therapy, a PD1 or PDL1 inhibitor, including Durvalumab or any anti-CTLA4 therapy, including Tremelimumab. 5. History of another primary malignancy except for: malignancy treated with curative intent and with no known active disease \>/= 5 years before the first dose of study drug and of low potential risk for recurrence; adequately treated non-melanoma skin cancer or lentigo maligna without evidence of disease; or adequately treated carcinoma in situ without evidence of disease, e.g., cervical cancer in situ. 6. Receipt of the last dose of anti-cancer therapy (chemotherapy, immunotherapy, endocrine therapy, targeted therapy, biologic therapy, tumor embolization, monoclonal antibodies, other investigational agent) \</= 28 days prior to the first dose of study drug (\</= 21 days prior to the first dose of study drug for subjects who have received prior TKIs \[e.g., erlotinib, gefitinib and crizotinib\] and \</= 6 weeks for nitrosourea, mitomycin C, or bevacizumab). (If sufficient wash-out time has not occurred due to the schedule or PK properties of an agent, a longer wash-out period may be required.) 7. Mean QT interval corrected for heart rate (QTc) \>/= 470 ms calculated from 3 electrocardiograms (ECGs) using Fridericia's Correction. 8. Current or prior use of immunosuppressive medication within 28 days before the first dose of Durvalumab OR Tremelimumab, with the exceptions of intranasal and inhaled corticosteroids or systemic corticosteroids at physiological doses, which are not to exceed 10 mg/day of prednisone, or an equivalent corticosteroid. Steroid premedication for the prevention of radiologic contrast hypersensitivity is allowed. 9. Any unresolved toxicity (\> CTCAE grade 1) from previous anti-cancer therapy, excluding alopecia. Subjects with irreversible toxicity greater than grade 1 that is not reasonably expected to be exacerbated by the investigational product may be included (e.g., hearing loss, peripherally neuropathy). 10. Any prior Grade \>/= 3 immune-related adverse event (irAE) while receiving any previous immunotherapy agent, or any unresolved irAE \> Grade 1. 11. Active or prior documented autoimmune disease within the past 2 years. NOTE: Subjects with vitiligo, Grave's disease, or psoriasis not requiring systemic treatment (within the past 2 years) are not excluded. 12. Active or prior documented inflammatory bowel disease (e.g., Crohn's disease, ulcerative colitis). 13. History of primary immunodeficiency. 14. History of allogeneic organ transplant. 15. History of hypersensitivity to Durvalumab, Tremelimumab, or any excipient. 16. Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, symptomatic congestive heart failure, uncontrolled hypertension, unstable angina pectoris, cardiac arrhythmia, active peptic ulcer disease or gastritis, active bleeding diatheses including any subject known to have evidence of acute or chronic hepatitis B, hepatitis C or human immunodeficiency virus (HIV), or psychiatric illness/social situations that would limit compliance with study requirements or compromise the ability of the subject to give written informed consent. 17. Known history of previous clinical diagnosis of tuberculosis. 18. History of leptomeningeal carcinomatosis or brain metastasis. 19. Unresolved partial or complete small or large bowel obstruction. 20. Receipt of live attenuated vaccination within 30 days prior to study entry or within 30 days of receiving Durvalumab OR Tremelimumab. 21. Subjects who are pregnant, breast-feeding or of reproductive potential who are not employing an effective method of birth control or are not willing to employ effective birth control from screening to 180 days after the last dose of Durvalumab + Tremelimumab combination therapy or 90 days after the last dose of Durvalumab monotherapy, whichever is the longer time period. 22. Any medical, social, or psychological condition that would interfere with evaluation of study treatment or interpretation of patient safety or study results. 23. Subjects with uncontrolled seizures. 24. Non-English speakers will be excluded from participating in the patient-reported outcomes component of the study.