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 </= 1.5 X ULN f. aspartate
             aminotransferase (AST/SGOT) and/or alanine aminotransferase (ALT/SGPT) </= 3 X ULN
             (</= 5 X ULN in patients with bone or liver metastases)

         12. Resolution of all acute toxic effects of prior therapy or surgical procedures to
             National Cancer Institute (NCI) CTCAE 4.03 Grade </= 1

         13. Women of child-bearing potential (intact uterus) MUST have a negative serum or urine
             human chorionic gonadotropin (HCG) test within 72 hours prior to receiving the first
             dose of study medication. Patients of childbearing potential are those who have not
             been surgically sterilized or have not been free from menses for > 1 year. If the
             urine test is positive or cannot be confirmed as negative, a serum pregnancy test will
             be required.

         14. Female patients of childbearing potential should be willing to use 2 methods of birth
             control or be surgically sterile, or abstain from heterosexual activity for the course
             of the study through 120 days after the last dose of study medication (Reference
             Section 7.7). Patients of childbearing potential are those who have not been
             surgically sterilized or have not been free from menses for > 1 year. Should a woman
             become pregnant or suspect she is pregnant while participating in this study, she
             should inform her treating physician immediately

         15. Pre/perimenopausal women must be amenable to be treated with goserelin. All patients
             will be rendered post-menopausal secondary to concomitant administration of goserelin.

         16. Ability to understand and willingness to sign informed consent form prior to
             initiation of the study and any study procedures.

        Exclusion Criteria:

          1. Patients who are pregnant or breastfeeding.

          2. The patient has serious preexisting medical condition(s) that would preclude
             participation in this study (for example, interstitial lung disease, severe dyspnea at
             rest or requiring oxygen therapy, history of major surgical resection involving the
             stomach or small bowel, or preexisting Crohn's disease or ulcerative colitis or a
             preexisting chronic condition resulting in baseline Grade 2 or higher diarrhea).

          3. Current use of food or drugs known to be potent CYP3A4 inhibitors, drugs known to be
             potent CYP3A4 inducers (for examples, see the Prohibited Concomitant Medications
             section), and drugs that are known to prolong the QT interval (see Prohibited
             Concomitant Medications in section 7.6.2).

          4. Diagnosis of another malignancy within 3 years, except for adequately treated basal
             cell or squamous cell skin cancer, or carcinoma in situ of the cervix.

          5. Previous chemotherapy or hormonal therapy for treatment of ovarian cancer.

          6. Known Hepatitis B, Hepatitis C or human immunodeficiency virus (HIV) infection.

          7. Inability or unwillingness to swallow pills.

          8. Active infection requiring intravenous (IV) antibiotics or other uncontrolled
             intercurrent illness requiring hospitalization.

          9. Inability to comply with the study and follow-up procedures.

         10. History of any of the following: syncope of cardiovascular etiology, ventricular
             arrhythmia of pathological origin (including, but not limited to, ventricular
             tachycardia and ventricular fibrillation), sudden cardiac arrest.

         11. Prior hematopoietic stem cell or bone marrow transplantation.

         12. Known abnormalities in coagulation such as bleeding diathesis, or treatment with
             anticoagulants precluding intramuscular injections of fulvestrant or goserelin (if
             applicable).

         13. Known or possible hypersensitivity to fulvestrant, or abemaciclib or any of their
             excipients.

         14. Pre/perimenopausal women with a known hypersensitivity to gnRH (gonadotropin-releasing
             hormone) agonists.