Inclusion Criteria:  Eastern Cooperative Oncology Group (ECOG) performance status 0-2. Histologically confirmed ovarian, fallopian tube or primary peritoneal cancer, high grade serous or high grade endometrioid histology subtype. Patients must have relapsed disease, either platinum-sensitive, resistant or refractory, with no limit to number of lines of prior systemic therapy. Radiographically documented disease progression within 28 days of registration. Patient must have measurable disease as per Response Evaluation Criteria in Solid Tumors (RECIST) 1.1 Progression on any prior Poly (ADP-ribose) polymerase (PARP) inhibitor therapy, with no limit to number of prior lines of PARP inhibitors. Patients must have adequate bone marrow, renal and hepatic function within 7 days of registration Left ventricular ejection fraction (LVEF) > 50% by echocardiograms or multigated acquisition (MUGA) scan within 28 days of registration. Patients are willing to undergo tumor biopsy pre-treatment (tissue at the time of progression on PARP inhibitor therapy). Availability of archival tissue (prior to PARP inhibitor therapy) for analysis. Women of child-bearing potential must agree to use a highly effective contraceptive method for study-required period. A negative high sensitive urine or serum pregnancy test within 3 days prior to the initiation of therapy will be required for women of childbearing potential. Patient must agree to not donate blood during the study or for 90 days after the last dose of study treatment. Patient must agree to not breastfeed during the study or for 30 days after the last dose of study treatment.  Exclusion Criteria:  Treatment with an investigational (other than PARP inhibitor) drug within 30 days and treatment with PARP inhibitor within 14 days prior to the first dose of study medication. Major surgery within 4 weeks of registration or ongoing clinically significant post-surgical complications. Patients with current or are at high-risk of developing fistula, or any other gastrointestinal disorders likely to interfere with absorption of the study medication. Patients with current or history of bowel obstruction within the last 3 months. Untreated unstable brain or leptomeningeal metastases. Greater than +1 proteinuria on two consecutive dipsticks within 14 days of registration. Unresolved toxicity of > grade 1 from previous anti-cancer therapy (including radiotherapy).  .History of poorly controlled hypertension or resting blood pressure >140/90 mmHg in the presence or absence of a stable regimen of anti-hypertensive therapy within 7 days of registration.  Mean QTc >470 msec in screening electrocardiograms within 7 days of registration or history of familial long QT syndrome. Any evidence of severe or uncontrolled diseases such as but not limited to unstable or uncompensated respiratory, cardiac, hepatic, renal disease or psychiatric illness/social situations that would limit compliance with study requirements. History of allergic reactions attributed to compounds of similar chemical or biologic composition to bevacizumab, paclitaxel, dostarlimab, or niraparib. Patients who have received prior weekly paclitaxel in the recurrent ovarian cancer setting. Patients who have received prior PD-1 inhibitor for ovarian cancer. Active autoimmune disease that has required systemic treatment in the past 2 years. History of interstitial lung disease. Patients with myelodysplastic syndrome/acute myeloid leukemia Previous allogenic bone marrow transplant. Immuno-compromised patients, e.g., patients who are known to be serologically positive for human immunodeficiency virus (HIV), patients with known active hepatitis (i.e., hepatitis B or C). Patients with significant hemorrhage (>30 mL bleeding/episode in previous 3 months) or hemoptysis (>5 mL fresh blood in previous 4 weeks). Patients who have had recent (within 2 weeks of registration, or until any wound has completely healed) major thoracic or abdominal surgery prior to study start, or a surgical incision that is not fully healed. History of stroke or transient ischemic attack within six months. Patients that are receiving other anti-cancer therapy (except patient currently progressing on treatment with PARP inhibitor), radiotherapy, biological therapy or other novel agent prior to start of study treatment. Other severe acute or chronic medical or psychiatric condition or laboratory abnormality that may increase the risk associated with study participation or study drug administration, or may interfere with the interpretation of study results, and in the judgment of the investigator would make the participant inappropriate for entry into the study. History of other primary second malignancies (except for adequately treated cutaneous basal or squamous cell carcinoma or carcinoma in situ) within < 3 years.