Fulvestrant Plus Abemaciclib in Women With Advanced Low Grade Serous Carcinoma

The goal of this clinical research study is to learn if fulvestrant and abemaciclib can help to control low-grade serous ovarian cancer. The safety of this drug combination will also be studied. This is an investigational study. Fulvestrant and abemaciclib are both FDA approved and commercially available for the treatment of several types of cancer. Their use in patients with low-grade serous ovarian cancer is investigational. The study doctor can explain how the study drugs are designed to work. Up to 15 participants will be enrolled in this study. All will take part at MD Anderson.

Inclusion Criteria: 1. Patients with clinical or surgical stage III or IV low-grade serous ovarian, primary peritoneal, or fallopian tube carcinomas who in the judgement of the treating physician are unlikely to achieve optimal surgical cytoreduction and have been recommended to receive neoadjuvant therapy. 2. Histological diagnosis must be based on surgical or core biopsy not just fine needle aspiration. Biopsies performed at other institutions must undergo pathology review and confirmation at MD Anderson Cancer Center. 3. Tissue from an archival tissue sample or fresh tissue obtained from a core or excisional biopsy of a tumor lesion. 4. Willingness to provide pre- and post-treatment tissue for translational studies. Pre-treatment fresh frozen tissue must be available for research purposes. This tissue can be collected from preoperative laparoscopy, other diagnostic biopsy, or a research-specific biopsy. 5. Signed informed consent on protocol LAB02-188. 6. Tissue from an archival tissue sample or fresh tissue obtained from a core or excisional biopsy of a tumor lesion. 7. Patients whose clinical biopsies are found to be insufficient for the planned translational studies must be willing to undergo a research biopsy. 8. Patients must have measurable disease by RECIST v1.1. a. Measurable disease is defined at least one lesion that can be accurately measured in at least one dimension (longest dimension to be recorded). Each target lesion must be \>20 mm when measured by conventional techniques, including palpation, plain x-ray, CT, and MRI, or \>10 mm when measured by spiral CT. 9. Women 18 years of age or older. 10. Eastern Cooperative Oncology Group (ECOG) performance status (PS) 0-1. 11. Abnormal organ function is permitted. However, patients must have: a. absolute neutrophil count \>/= 1500/mL b. platelets \>/= 100,000/mL c. hemoglobin \>/= 9 g/dL d. estimated creatinine clearance \>/= 60 ml/min as calculated using the method standard for the institution e. total serum bilirubin \