A Phase II Single Arm Pilot Study of the Chk1/2 Inhibitor (LY2606368) in BRCA1/2 Mutation Associated Breast or Ovarian Cancer, Triple Negative Breast Cancer, High Grade Serous Ovarian Cancer, and Metastatic Castrate-Resistant Prostate Cancer

Background: * Checkpoint kinases 1 and 2 (Chk1/2) are major regulators of the cell cycle and are intimately associated with the cellular response to deoxyribonucleic acid (DNA) damage and repair. Chk1/2 also function as the primary mediators of cell cycle arrest in tumors with tumor protein P53 (p53) dysfunction, such as high-grade serous ovarian cancer (HGSOC), and triple negative breast cancer (TNBC). * Participants with germline BRCA1 BReast CAncer gene 1 and BRCA2 BReast CAncer gene 2 (BRCA1/2) mutation have inherent defects in DNA damage repair pathways. * Chk1/2 inhibition alone yielded DNA damage and mitotic catastrophe preclinically, even in the absence of DNA damage by external agents in tumors with underlying DNA repair dysfunction. * The second-generation Chk1/2 inhibitor (Prexasertib (LY2606368) yielded safety and preliminary single agent activity in advanced cancer participants. * We hypothesize that LY2606368 will result in clinical benefit in participants with Germline BRCA-Mutated (gBRCAm)-associated breast or ovarian cancers, and HGSOC and TNBC with low genetic risk. Objectives: * To determine the objective response rate (Complete Response (CR)+Partial Response (PR) of single agent LY2606368 in patients with gBRCAm-associated breast or ovarian cancer, HGSOC and TNBC with low genetic risk. * To determine the safety and toxicity, and progression-free interval (PFI) of LY2606368 in pretreated participants. * To determine biochemical changes in the DNA damage repair and cell cycle check point pathways in tumor and blood samples in response to treatment. * To determine potential resistance mechanisms to LY2606368 treatment in HGSOC. Eligibility: * Participants with recurrent/refractory BReast CAncer gene (BRCA) mutant breast or ovarian cancer, HGSOC, and TNBC, for whom there remains no standard curative measures. * A documented deleterious germline or somatic BRCA mutation for breast or ovarian cancer participants enrolling in Cohort 1. * Negative BRCA mutation testing or negative family history of hereditary breast and ovarian cancer syndrome for HGSOC (Cohort 2). * Negative BRCA mutation testing or negative family history of hereditary breast and ovarian cancer syndrome for TNBC (Cohort 3). * Effective with amendment I (version date 4/24/2017), Metastatic Castration-Resistant Prostate Cancer (mCRPC), Cohort 4 was closed. * Negative BRCA mutation testing or negative family history of hereditary breast and ovarian cancer syndrome for recurrent platinum-resistant HGSOC with measurable and biopsiable disease (Cohort 5). * Negative BRCA mutation testing or negative family history of hereditary breast and ovarian cancer syndrome for recurrent platinum-resistant HGSOC with measurable but without biopsiable disease (Cohort 6). * Participants must be off prior chemotherapy, radiation therapy, hormonal therapy, or biological therapy for at least 4 weeks. * Eastern Cooperative Oncology Group (ECOG) performance status 0-2 and adequate organ and marrow function. Design: * This is an open label, single arm phase II trial to examine activity of LY2606368 in participants in the 6 independent cohorts (Cohorts 1-6). * LY2606368 will be dosed at the recommended phase 2 dose (RP2D) of 105 mg/m\^2 intravenous (IV) once every 14 days of a 28 day-cycle. * Research samples including whole blood, circulating tumor cells (CTCs), and tumor biopsies will be obtained for pharmacodynamics (PD) endpoints at baseline, Cycle 1 Day 15 (6-24hour (hr) post-2nd dose), and/or at progression in all participants. Tumor biopsies will not be performed in Cohort 6. * Participants (Cohorts 1-3, 5 and 6) will be evaluated every two cycles for response using Response Evaluation Criteria in Solid Tumors (RECIST) v1.1 and every cycle for safety using Common Terminology Criteria for Adverse Events (CTCAE) v4.0.