FT516 and IL2 With Enoblituzumab for Ovarian Cancer

Inclusion Criteria: * Recurrent epithelial ovarian cancer, fallopian tube, or primary peritoneal cancer meeting one of the following minimal prior treatment requirement (no limit to the maximum number of prior treatments): * Platinum Resistant: may receive FT516 as 2nd line (as 1st salvage therapy) with platinum resistant is defined as disease that has responded to initial chemotherapy but demonstrates recurrence within a relatively short period of time (\< 6 months) following the completion of treatment. * Platinum Sensitive: may receive FT516 as 3rd line therapy (as 2nd salvage therapy) with platinum sensitive is defined as the recurrence of active disease in a patient who has achieved a documented response to initial platinum-based treatment and has been off therapy for an extended period of time (≥ 6 months). * Measurable disease per modified Response Evaluation Criteria in Solid Tumors, v1.1 within the abdomen and pelvis assess within 42 days of the 1st FT516 infusion. Extra-peritoneal disease is permitted; however each lesion must be \< 5 cm at the largest diameter. * At least 18 years of age * GOG Performance Status 0, 1, or 2 * Adequate organ function within 14 days of study registration (28 days for pulmonary and cardiac) defined as: * Hematologic: platelets ≥ 75,000 x 10\^9/L and hemoglobin ≥ 9 g/dL, unsupported by transfusions; absolute neutrophil count (ANC) ≥ 1000 x 10\^9/L, unsupported by G-CSF or granulocytes * Creatinine: Estimated glomerular filtration rate (eGFR) ≥ 60 mL/min/1.73m\^2 per current institutional calculation formula * Hepatic: AST and ALT ≤ 3 x upper limit of institutional normal * Pulmonary Function: Oxygen saturation ≥ 90% on room air; PFT's required only if symptomatic or prior known impairment - must have pulmonary function \>50% corrected DLCO and FEV1 * Cardiac Function: LVEF ≥ 40% by echocardiography, MUGA, or cardiac MRI; no clinically significant cardiovascular disease including any of the following: stroke or myocardial infarction within 6 months prior to first study treatment; unstable angina or congestive heart failure of New York Heart Association Grade 2 or higher * Agrees to the placement of an intraperitoneal catheter before the 1st dose of study directed drug (chemotherapy or enoblituzumab - Cohort 4 and 5) and remains in place through Day 36 or longer if retreatment is planned * Agrees to undergo a tumor biopsy if feasible at the time the catheter is placed and removed - Accessible tumor for biopsy is not required for eligibility. * Washout period of at least 14 days after any standard of care tumor directed therapy prior to the first dose of investigational product (FT516 for Levels 1-3 or enoblituzumab for Levels 4-5) * If history of brain metastases must be stable for at least 3 months after treatment - A brain CT scan or MRI is only be required in subjects with known brain metastases at the time of enrollment or in subjects with clinical signs or symptoms suggestive of brain metastases * Must agree to and sign the consent for the companion Long-Term Follow-Up study (CPRC #2020LS072) to fulfill the FDA required 15 years of follow-up for a genetically modified cell product * Voluntary written consent prior to the performance of any research related procedures Exclusion Criteria: * Pregnant or breastfeeding or planning on becoming pregnant in the next 6 months. Woman of childbearing potential who still have a uterus and ovaries, must agree to use at effective contraception and must have a negative pregnancy test within 14 days of study enrollment. * Any known condition that requires systemic immunosuppressive therapy (\> 5mg prednisone daily or equivalent) during the FT516 dosing period (3 days before the 1st dose through 14 days after the last dose) - topical and inhaled steroids are permitted. * Active autoimmune disease requiring systemic immunosuppressive therapy * History of severe asthma and currently on chronic systemic medications (mild asthma requiring inhaled steroids only is eligible) * Uncontrolled bacterial, fungal or viral infections with progression of clinical symptoms despite therapy * Receipt of any investigational agent within 28 days prior to the first dose of investigational product (FT516 for Levels 1-3 or enoblituzumab for Levels 4-5) * Live vaccine \<6 weeks prior to start of lympho-conditioning * Known allergy to the following FT516 components: albumin (human) or DMSO * Any history of prior enoblituzumab administration * Known history of HIV positivity or active hepatitis C or B - chronic asymptomatic viral hepatitis is allowed * Presence of any medical or social issues that are likely to interfere with study conduct or may cause increased risk to patient