Paclitaxel, Pembrolizumab and Olaparib in Previously Treated Advanced Gastric Adenocarcinoma

Inclusion Criteria: * Age ≥18 years. * Eastern Cooperative Oncology Group (ECOG) performance status 0 or 1 * Must have advanced gastric or gastroesophageal cancer. * Must have received only one prior line of systemic therapy for advanced disease and experienced disease progression on this regimen.. * For Cohort 1: Must have the presence of measurable lesion. * Must agree to have a biopsy. * Life expectancy of greater than 3 months. * Patients must have adequate organ and marrow function defined by study - specified laboratory tests. * Woman of childbearing potential must have a negative pregnancy test. * Must use acceptable form of birth control while on study. * Ability to understand and willingness to sign a written informed consent document. Exclusion Criteria: * Known active central nervous system (CNS) metastases and/or carcinomatous meningitis. Require any antineoplastic therapy. * Require any other form of systemic or localized antineoplastic therapy. * Has received prior therapy with paclitaxel or PARP inhibitor. Previous paclitaxel may be allowed if no progression on or within 6 months of receiving this drug. * Hypersensitivity reaction to any paclitaxel, pembrolizumab or related compounds and/or to any of the components. * Allergy to dexamethasone, diphenhydramine and famotidine. * Is taking a moderate or strong CYP3A inhibitor. * Has uncontrolled intercurrent acute or chronic medical illness. * Has a known additional malignancy that is progressing and has required active treatment within the past 1 year. * Has received prior systemic anti-cancer therapy including investigational agents within 2 weeks prior to study treatment. * Has received prior radiotherapy within 2 weeks of start of study treatment. * Has received a live vaccine within 30 days prior to the first dose of study drug. * Is currently or has participated in another investigational study within 4 weeks prior to receiving study drug. * Has an active known or suspected autoimmune disease. * Has a diagnosis of immunodeficiency. * Prior tissue or organ allograft or allogeneic bone marrow transplantation. * Has a history or current evidence of any condition, therapy, or laboratory abnormality that might confound the results of the study, interfere with the subject's participation for the full duration of the study, or is not in the best interest of the subject to participate, in the opinion of the treating investigator. . * Requires daily supplemental oxygen. * History of (non-infectious) pneumonitis that required steroids or has current pneumonitis. * History of encephalitis, meningitis, or uncontrolled seizures in the year prior to informed consent. * Infection with HIV or hepatitis B or C at screening. * Has uncontrolled infection requiring systemic therapy.. * Subjects unable to undergo venipuncture and/or tolerate venous access. * Has known psychiatric or substance abuse disorder that would interfere with cooperation with the requirements of the trial. * Woman who are pregnant or breastfeeding. * A woman of childbearing potential (WOCBP) who has a positive urine pregnancy test within 72 hours prior to study drug initiation.