Adavosertib With or Without Olaparib in Treating Patients With Recurrent Ovarian, Primary Peritoneal, or Fallopian Tube Cancer

Inclusion Criteria: 1. Has read and understands the informed consent form (ICF) and has given written informed consent prior to any study procedures. 2. Histologically confirmed recurrent epithelial ovarian, primary peritoneal, or fallopian tube cancer for which there is no known or established treatment available with curative intent. 3. Have demonstrated progressive disease while taking a licensed PARP inhibitor as a previous therapy or within 6 months of completing PARP inhibitor therapy. Response to prior PARPi is not required. However, there must be documented evidence of progression prior to study entry. 4. Prior PARP therapy could have been administered as either treatment for recurrent disease or as maintenance following prior treatment. 5. At least one measurable lesion according to RECIST v1.1. 6. Adequate archived primary or metastatic tumor tissue collected before the prior PARP therapy. 7. Eastern Cooperative Oncology Group (ECOG) Performance Status (PS) score of 0-1 within 28 days of study entry. 8. Patients must have adequate organ and bone marrow function measured within 14 days of study drug(s) initiation: 1. Absolute neutrophil count (ANC) ≥1500/μL 2. Hemoglobin (Hgb) ≥10 g/dL with no blood transfusion in the past 28 days 3. Platelets ≥100,000/μL with no platelet transfusion in the past 28 days 4. Alanine Transaminase (ALT) and Aspartate Transaminase (AST) ≤2.5 x upper limit of normal (ULN) or ≤5 x ULN if known hepatic metastases. 5. Serum bilirubin within normal limits (WNL) or ≤1.5 x ULN in patients with liver metastases; or total bilirubin ≤3.0 x ULN with direct bilirubin WNL in patients with well documented Gilbert's Syndrome. 6. Patients must have creatinine clearance (CrCl) of ≥51 mL/min estimated or measured using standard methodology at the investigating centre (i.e. Cockcroft- Gault, or 24 hr urine): Estimated CrCl = (140-age \[years\]) x weight (kg) (x F)a serum creatinine (mg/dL) x 72 a where F=0.85 for females and F=1 for males 9. Women who are not of child-bearing potential and fertile females of childbearing potential who agree to use adequate contraceptive measures, who are not breastfeeding, and who have a negative serum pregnancy test within 3 days prior to the start of study treatment (see Appendix E).Predicted life expectancy ≥16 weeks. 10. Must be ≥18 years of age. 11. Willingness and ability to comply with study and follow-up procedures. Inclusion criteria for ceralasertib arm only: 1. Must have platinum-sensitive disease, defined as no clinical or radiographic evidence of disease recurrence for \> 6 months (or 180 days) after last receipt of platinum- based therapy. Patients must have had response (complete or partial) to their prior line of platinum therapy and cannot have had progression through prior platinum-based therapy. 2. Have demonstrated progressive disease while taking a licensed PARP inhibitor as a previous therapy, used in the maintenance or treatment setting. 3. Are germline or somatic BRCA mutant or HRD positive by a licensed test prior to enrollment. 4. Patient must have demonstrated clinical benefit from prior PARPi treatment. Clinical benefit after \>1 line of chemotherapy is defined as a minimum duration of continuous treatment with the PARP inhibitor of 6 months in the treatment or maintenance setting (which may or may not be associated with response by CA-125 or imaging). If the PARP inhibitor is received as maintenance following first-line chemotherapy, the minimum duration of continuous treatment with the PARP inhibitor is 12 months 5. Patients who discontinue the PARP inhibitor for any reasons other than progression are ineligible 6. If the prior PARP inhibitor used was olaparib, then patients must have received treatment without significant toxicity or the need for a permanent dose reduction. 7. Documented evidence of progression prior to study entry. Exclusion criteria: 1. Prior Treatment: 1. PARP inhibitor: Most oral PARP inhibitors (the immediate prior therapy) have a half- life for which 5 half-lives is ≤21 days. Thus, a minimum of 10 days between termination of the prior treatment and administration of olaparib and/or AZD1775 treatment is required. In the event a PARP inhibitor has a longer half-life where 5 half-lives are ≥ 21 days, treatment of olaparib and/or AZD1775 should not begin for 5 half-lives or at least 21 days, whichever is shorter. 2. Any hormonal therapy directed at the malignant tumor must be discontinued at least one week prior to registration (study enrollment). Continuation of hormone replacement therapy is permitted. 3. Any other prior therapy directed at the malignant tumor, including any systemic immunologic agents, must be discontinued at least three weeks prior to first dose of study drug (6 weeks for nitrosoureas or mitomycin C). Cytotoxic chemotherapy, hormonal or non-hormonal targeted therapy within 21 days of Cycle 1 Day 1 is not permitted (a duration of 30 days or 5 half-lives (whichever is shorter) is required for patients treated with non-cytotoxic drugs). The minimum washout period for immunotherapy is 42 days. Palliative radiotherapy must have been completed 21or more days before Cycle 1 Day 1 (with the exception of patients receiving radiation to more than 30% of the bone marrow or with a wide field of radiation within 4 weeks of the first dose of study treatment). The patient can receive a stable dose of bisphosphonates or denosumab for bone metastases, before and during the study as long as these were started at least 5 days prior to the study treatment. Receiving, or having received during the 14 days prior to first dose, corticosteroids (at a dose \> 10 mg prednisone/day or equivalent) for any reason. 2. Major surgical procedures ≤28 days of beginning study treatment, or minor surgical procedures ≤7 days. No waiting period required following port-a-cath placement. 3. Grade \>1 toxicity from prior therapy (except alopecia or anorexia). 4. Patient has an inability to swallow oral medications and patients with gastrointestinal disorders likely to interfere with absorption of the study medication. Note: Patient may not have a percutaneous endoscopic gastrostomy (PEG) tube or be receiving total parenteral nutrition (TPN). Patient has refractory nausea and vomiting, chronic gastrointestinal diseases or previous significant bowel resection, with clinically significant sequelae that would preclude adequate absorption of the study medication. 5. Patients with symptomatic uncontrolled brain metastases. Patients with a history of treated central nervous system (CNS) metastases are eligible provided they meet all of the following criteria: disease outside the CNS is present, no clinical evidence of progression since completion of CNS-directed therapy, minimum 3 weeks between completion of radiotherapy and Cycle 1 Day 1 and recovery from significant (Grade ≥ 3) acute toxicity with no ongoing requirement for \>10 mg of prednisolone per day or an equivalent dose of other corticosteroid. If on corticosteroids, the patient should be receiving a stable dose of corticosteroids, started at least 4 weeks prior to treatment. 6. Patient has had a prescription or non-prescription drugs or other products known to be sensitive to CYP3A4 substrates or CYP3A4 substrates with a narrow therapeutic index, or to be moderate to strong inhibitors/inducers of CYP3A4 which cannot be discontinued 2 weeks prior to Day -3 of dosing and withheld throughout the study until 2 weeks after the last dose of study drug. Co- administration of aprepitant or fosaprepitant during this study is prohibited (see Appendix G). 7. The use of herbal supplements or 'folk remedies' (and medications and foods that significantly modulate CYP3A activity) should be discouraged. These herbal medications include but are not limited to: St. John's wort, kava, ephedra (mahung), gingko biloba, dehydroepiandrosterone (DHEA), yohimbe, saw palmetto and ginseng. Patients should stop using herbal medications 7 days prior to first dose of study treatment. If deemed necessary, such products may be administered with caution and the reason for use documented in the CRF. 8. Any known hypersensitivity or contraindication to the components of the study drugs AZD1775, ceralasertib or olaparib. 9. Any of the following cardiac diseases currently or within the last 6 months as defined by New York Heart Association (NYHA) ≥ Class 2 (see Appendix F). 1. Unstable angina pectoris 2. Congestive heart failure 3. Acute myocardial infarction 4. Conduction abnormality not controlled with pacemaker or medication 5. Significant ventricular or supraventricular arrhythmias (patients with chronic rate- controlled atrial fibrillation in the absence of other cardiac abnormalities are eligible) 10. AZD1775 should not be given to patients who have a history of Torsades de pointes unless all risk factors that contributed to Torsades have been corrected. AZD1775 has not been studied in patients with ventricular arrhythmias or recent myocardial infarction. 11. Participants with a mean resting corrected QT interval (QTc) ≥ 480msec at study entry, as calculated by the Frederica formula (QTcF) by institutional standards obtained from an electrocardiogram (ECG) or congenital long QT syndrome. (Note: if one ECG demonstrates a QTcF \>480 msec, then a mean QTcF of ≤ 480 msec obtained from 3 ECGs 2-5 minutes apart is required at study entry. 12. Any factors that increase the risk of QTc prolongation or risk of arrhythmic events such as congestive heart failure, unstable angina pectoris, acute myocardial infarction, hypokalaemia, congenital long QT syndrome, immediate family history of long QT syndrome or unexplained sudden death under 40 years of age, conduction abnormality not controlled with pacemaker or medication. 13. Pregnant or breast-feeding. 14. Serious active infection at the time of study entry, or another serious underlying medical condition that would impair the ability of the patient to receive study treatment. Examples include, but are not limited to, active bleeding diatheses, renal transplant, uncontrolled major seizure disorder, severe COPD, superior vena cava syndrome, extensive bilateral lung disease on High Resolution CT scan, severe Parkinson's disease, active inflammatory bowel disease, psychiatric condition, or active infection including any patient known to have hepatitis B, hepatitis C and human immunodeficiency virus (HIV) or requiring systemic antibiotics, antifungals or antiviral drugs. Screening for chronic conditions is not required 15. Presence of other active invasive cancers. 16. Psychological, familial, sociological, or geographical conditions that do not permit compliance with protocol 17. Patients with myelodysplastic syndrome/acute myeloid leukemia or with features suggestive of MDS/AML. 18. Previous allogenic bone marrow transplant or double umbilical cord blood transplantation (dUCBT). Exclusion criteria for ceralasertib arm only: 1. Concomitant use of known potent cytochrome P (CYP) 3A inhibitors (e.g., itraconazole, telithromycin, clarithromycin, protease inhibitors boosted with ritonavir or cobicistat, indinavir, saquinavir, nelfinavir, boceprevir, telaprevir) or moderate CYP3A inhibitors (e.g., ciprofloxacin, erythromycin, diltiazem, fluconazole, verapamil). The required washout period prior to starting study treatment is 2 weeks. 2. Concomitant use of known strong (e.g., phenobarbital, enzalutamide, phenytoin, rifampicin, rifabutin, rifapentine, carbamazepine, nevirapine and St John's Wort) or moderate CYP3A inducers (e.g., bosentan, efavirenz, modafinil). The required washout period prior to starting study treatment is 5 weeks for enzalutamide or phenobarbital and 3 weeks for other agents. 3. Patient has had prescription or non-prescription drugs or other products known to be CYP3A4 and/or CYP2B6 substrates or CYP3A4 and/or CYP2B6 substrates with a narrow therapeutic index. Exposure of other drugs metabolized by CYP3A4 and/or CYP2B6 may be reduced and additional monitoring may be required 4. Patients at risk of brain perfusion problems, e.g., medical history of carotid stenosis or pre-syncopal or syncopal episodes, history of TIAs 5. Uncontrolled hypertension (grade 2 or above) requiring clinical intervention 6. A diagnosis of ataxia telangiectasia 7. Any other malignancy which has been active or treated within the past three years, with the exception of cervical intra-epithelial neoplasia and non-melanoma skin cancer, Ductal Carcinoma curatively treated with non-evidence of disease for ≥ 5 years prior to study entry 8. With the exception of alopecia and CTCAE grade 2 neuropathy, any unresolved toxicities from prior therapy ≥ Common Terminology Criteria for Adverse Events (CTCAE) grade 2 9. Patients with relative hypotension (\<90/60 mm Hg) or clinically relevant orthostatic hypotension, including a fall in blood pressure of \> 20 mm Hg 10. Non-leukocyte depleted whole blood transfusion within 120 days of the date of the genetic sample collection