Dose-Escalation Study of Intraperitoneal (IP) Cisplatin, IV/IP Paclitaxel, IV Bevacizumab, and Oral Olaparib for Newly Diagnosed Ovarian, Primary Peritoneal, and Fallopian Tube Cancer

Inclusion Criteria: * Subjects with histologic diagnosis of epithelial ovarian carcinoma, primary peritoneal carcinoma, or fallopian tube carcinoma, following initial surgery. All subjects must have had appropriate surgery for ovarian, primary peritoneal, or fallopian tube carcinoma with appropriate tissue available for histologic evaluation to confirm diagnosis and stage. Pathology must be verified at Memorial Sloan-Kettering Cancer Center. Patients with initial cytoreduction surgery performed at outside hospitals will be eligible for this protocol. * Patients with the following histologic cell types are eligible: * High grade serous adenocarcinoma * Endometrioid adenocarcinoma * Undifferentiated carcinoma * Clear cell adenocarcinoma * Mixed epithelial adenocarcinoma * Adenocarcinoma not otherwise specified (N.O.S.) * Carcinosarcoma * Subjects must have a Karnofsky Performance Status (KPS) of ≥ 70%. * Subjects must be entered no more than 12 weeks postoperatively. * Subjects must agree to undergo genetic counseling and BRCA testing. Patients in the expansion cohort must have a germline BRCA 1 or 2 mutation. * Physical and Laboratory Test Findings * PT such that international normalized ratio (INR) is ≤1.5 (or an in-range INR, usually between 2 and 3, if a patient is on a stable dose of therapeutic warfarin) and a PTT ≤ 1.5 times the upper limit of normal. * Bone marrow function: * Absolute neutrophil count (ANC) ≥ 1,500/mcL * Platelets ≥ 100,000/mcL * Renal function: * Creatinine ≤ 1.5 mg/dl * Hepatic function: * Bilirubin ≤ 1.5 x ULN * AST and ALT ≤ 3 x ULN * Neurologic function: * Neuropathy (sensory) ≤ CTC Grade 1 * Urine Protein Creatinine: * Urine protein creatinine (UPC) ratio must be \< 1.0 gm. * If UPC ratio ≥1, collection of 24-hour urine measurement of urine protein is recommended \* * UPC ratio of spot urine is an estimation of the 24 urine protein excretion - a UPC ratio of 1 is roughly equivalent to a 24-hour urine protein of 1 gm. UPC ratio is calculated using one of the following formulas: 1. \[urine protein\]/\[urine creatinine\] - if both protein and creatinine are reported in mg/dL 2. \[(urine protein) x0.088\]/\[urine creatinine\] - if urine creatinine is reported in mmol/L \*The UPCR has been found to correlate directly with the amount of protein excreted in a 24 hour urine collection. Specifically, a UPCR of 1.0 is equivalent to 1.0 gram of protein in a 24 hour urine collection. Obtain at least 4 ml of a random urine sample in a sterile container (does not have to be a 24 hour urine). Send sample to lab with request for urine protein and creatinine levels \[separate requests\]. The lab will measure protein concentration (mg/dL) and creatinine concentration (mg/dL). The UPCR is derived as follows: protein concentration (mg/dL)/creatinine (mg/dL). Patients must have a UPCR \< 1.0 to allow participation in the study. * Patients of childbearing potential must have a negative serum pregnancy test prior to study entry and be practicing an effective form of contraception during the study and for at least 6 months after receiving the final treatment of bevacizumab. * Patients must have an Intraperitoneal (IP) port in place. If a patient does not have an IP port, she must be willing to undergo surgical placement of one. Exclusion Criteria: * Subjects with a current diagnosis of epithelial ovarian tumor of low malignant potential (borderline carcinomas) are not eligible. * Patients with synchronous primary endometrial cancer or a past history of endometrial cancer, unless all of the following conditions are met: * Stage not greater than IB * No more than superficial myometrial invasion * No vascular or lymphatic invasion * No poorly differentiated subtypes, including papillary serous, clear cell, or other FIGO Grade 3 lesions. * Subjects who have received prior radiotherapy to any portion of the abdominal cavity or pelvis are excluded. Prior intravaginal brachytherapy is permitted. Prior radiation for localized cancer of the breast, head and neck, or skin is permitted, provided that it was completed more than 3 years prior to enrollment, and the subject remains free of recurrent or metastatic disease. * Patients who have received prior chemotherapy for any abdominal or pelvic tumor are excluded. Patients who have received neoadjuvant chemotherapy prior to their initial debulking are excluded. Patients may have received prior adjuvant chemotherapy for localized breast cancer. * With the exception of non-melanoma skin cancer and other specific malignancies as noted above, subjects with other invasive malignancies who had (or have) any evidence of the other cancer present within the last 3 years or whose previous cancer treatment contraindicates this protocol therapy are excluded. * Patients who have received prior bevacizumab (or any other VEGF targeted agent) or prior PARP inhibitor. * Subjects with acute hepatitis. * Subjects with active infection that requires parenteral antibiotics. * \> Grade 1 hearing loss or tinnitus * Patients who are pregnant or nursing. * Patients under the age of 18. * Patients with clinical symptoms or signs of gastrointestinal obstruction, require parenteral hydration or nutrition, require a drainage gastrostomy tube and/or have any other impairment that limits their ability to take oral medication. * Patients with serious non-healing wound, ulcer, or bone fracture including history of abdominal fistula, gastrointestinal perforation or intra-abdominal abscess within 28 days. Patients with granulating incisions healing by secondary intention with no evidence of fascial dehiscence or infection are eligible but require weekly wound examinations. * Patients with active bleeding or pathologic conditions that carry high risk of bleeding, such as known bleeding disorder, coagulopathy, or tumor involving major vessels. * Patients with history or evidence upon physical examination of CNS disease, including primary brain tumor, seizures not controlled with standard medical therapy, any brain metastases, or history of cerebrovascular accident (CVA, stroke), transient ischemic attack (TIA) or subarachnoid hemorrhage within six months of the first date of treatment on this study. * Patients with clinically significant cardiovascular disease including * Uncontrolled hypertension, defined as systolic \> 150 mm Hg or diastolic \> 90 mm Hg. * Myocardial infarction or unstable angina \< 6 months prior to registration. * New York Heart Association (NYHA) Class II or higher congestive heart failure * Serious cardiac arrhythmia requiring medication. * CTEP CTCAE Version 4.0, Grade 2 or higher peripheral ischemia \[brief (\<24 hrs) episode of ischemia managed non-surgically and without permanent deficit\]. * Core biopsy or other minor surgical procedure (excluding placement of a vascular access device, paracentesis, and/or thoracentesis) within 7 days prior to the first date of bevacizumab therapy. * Patients with known hypersensitivity to Chinese hamster ovary cell products or other recombinant human or humanized antibodies. * Patients receiving any medications or substances that are strong inhibitors or inducers of CYP3A4. Lists including medications or substances known or with the potential to interact with these enzymes.