A Phase III Study Comparing Single-Agent Olaparib or the Combination of Cediranib and Olaparib to Standard Platinum-Based Chemotherapy in Women With Recurrent Platinum-Sensitive Ovarian, Fallopian Tube, or Primary Peritoneal Cancer

• Patients must have platinum-sensitive recurrent high-grade serous or high-grade endometrioid ovarian, primary peritoneal, or fallopian tube cancers; patients with other (clear cell, mixed epithelial, undifferentiated carcinoma, or transitional cell carcinoma) high-risk histologies are also eligible, provided that the patient has a known deleterious germline BRCA1 or BRCA2 mutation identified through testing at a clinical laboratory; Note: Due to the long acceptance of germline BRCA testing through Myriad, Myriad testing will be accepted; if testing for germline BRCA is done by other organizations, documentation from a qualified medical professional (e.g., ovarian cancer specialty physician involved in the field, high risk genetics physician, genetics counselor) listing the mutation and confirming that the laboratory results showed a recognized germline deleterious BRCA1 or BRCA2 mutation or BRCA rearrangement is required; please collect a copy of Myriad or other BRCA mutational analysis (positive or VUS or negative) reports

‣ Platinum-sensitive disease defined as no clinical or radiographic evidence of disease recurrence for \> 6 months (or 182 days) after last receipt of platinum-based therapy

⁃ Patients must have had a complete clinical response to their prior line of platinum therapy and cannot have had progression through prior platinum-based therapy

• Patients must have signed an approved informed consent and authorization permitting release of personal health information

• Patients must have evaluable disease - defined as one of the following:

‣ Response Evaluation Criteria in Solid Tumors (RECIST) 1.1 measurable disease OR

⁃ Evaluable disease (defined as solid and/or cystic abnormalities on radiographic imaging that do not meet RECIST 1.1 definitions for target lesions OR ascites and/or pleural effusion that has been pathologically demonstrated to be disease-related) AND a cancer antigen 125 (CA125) that has doubled from the post-treatment nadir and is also greater than 2 times upper limit of normal (ULN)

• Prior therapy:

‣ Prior chemotherapy must have included a first-line platinum-based regimen with or without intravenous consolidation chemotherapy

⁃ Patients may have received an unlimited number of platinum-based therapies in the recurrent setting

⁃ Patients may have received up to 1 non-platinum-based line of therapy in the recurrent setting; prior hormonal therapy will not be considered to count as this non-platinum-based line

⁃ Patients may not have had a prior anti-angiogenic agent in the recurrent setting; prior use of bevacizumab in the upfront or upfront maintenance setting is allowed

⁃ Patients may not have previously received a poly adenosine diphosphate (ADP) ribose polymerase (PARP)-inhibitor

⁃ Prior hormonal-based therapy for ovarian, primary peritoneal, or fallopian tube cancer is acceptable

• Patients must have an Eastern Cooperative Oncology Group (ECOG) performance status of 0, 1 or 2 (Karnofsky \>= 60%)

• Absolute neutrophil count \>= 1,500/mcL

• Platelets \>= 100,000/mcL

• Hemoglobin \>= 10 g/dL

• Creatinine =\< the institutional upper limit of normal (ULN) OR creatinine clearance \>= 60 mL/min/1.73 m\^2 for patients with creatinine levels above institutional normal

• Urine protein: creatinine ratio (UPC) of =\< 1 or less than or equal to 2+ proteinuria on two consecutive dipsticks taken no less than 1 week apart; UPC is the preferred test; patients with \>= 2+ proteinuria on dipstick must also have a 24 hour urine collection demonstrating =\< 500 mg over 24 hours

• Total bilirubin =\< 1.5 x the institutional ULN

• Aspartate aminotransferase (AST) (serum glutamic oxaloacetic transaminase \[SGOT\]) and alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SGPT\]) =\< 3 times institutional ULN

• Toxicities of prior therapy (excepting alopecia) should be resolved to less than or equal to grade 1 as per National Cancer Institute-Common Terminology Criteria for Adverse Events (NCI-CTCAE); patients with long-standing stable grade 2 neuropathy may be considered after discussion with the overall principal investigator (PI), but may not receive carboplatin and paclitaxel as the reference regimen, if randomized to that arm

• Patients must be able to swallow and retain oral medications and without gastrointestinal illnesses that would preclude absorption of cediranib or olaparib

• Patients must have adequately controlled blood pressure (BP), with a BP no greater than 140 mmHg (systolic) and 90 mmHg (diastolic) for eligibility; patients must have a BP of =\< 140/90 mmHg taken in the clinic setting by a medical professional within 2 weeks prior to starting study; patients with hypertension may be managed with up to a maximum of three antihypertensive medications; it is strongly recommended that patients who are on three antihypertensive medications be followed by a cardiologist or blood pressure specialist for management of blood pressure while on protocol

• Patients must be willing and able to check and record daily blood pressure readings; blood pressure cuffs will be provided to patients randomized to Arm III

• Cediranib has been shown to terminate fetal development in the rat, as expected for a process dependent on VEGF signaling; for this reason, women of child-bearing potential must have a negative pregnancy test prior to study entry; women of child-bearing potential must agree to use two reliable forms of contraception (hormonal or barrier method of birth control; abstinence) prior to study entry, for the duration of study participation, and for 6 weeks after cediranib discontinuation; should a woman become pregnant or suspect she is pregnant while participating in this study, she should inform her treating physician immediately

• Adequately controlled thyroid function, with no symptoms of thyroid dysfunction and thyroid stimulating hormone (TSH) within normal limits

• Age \>= 18