Inclusion Criteria:

          1. Subjects must have hepatocellular carcinoma (HCC) that is inoperable (where surgery is
             not indicated due to disease extent, co-morbidities, or other technical reasons)

          2. Histologic confirmation of HCC is not required for screening but is required prior to
             initiation of study treatment. Subjects with hepatocholangiocarcinoma or
             cholangiocarcinoma are not eligible.

          3. Tumor must be positive for retinoblastoma (RB) expression by immunohistochemistry

          4. Age > 18 years and ability to understand and the willingness to sign a written
             informed consent document.

          5. ECOG performance status of 0 or 1

          6. Childs-Pugh score of <7

          7. Life expectancy of at least 12 weeks

          8. Must be able to swallow tablets

          9. Must be willing to comply with protocol procedures (including completion of diaries
             and outcome measures)

         10. Local or loco-regional therapy to the liver (i.e. surgery, radiation therapy, hepatic
             arterial embolization, chemoembolization, radiofrequency ablation, percutaneous
             ethanol injection, or cryoablation) must have been completed > 4 weeks prior to
             enrollment

         11. Must be willing to undergo a pretreatment and on-treatment biopsy and have a tumor
             site that is accessible for core needle biopsy

         12. Measurable or evaluable disease as defined by RECIST v. 1.1

         13. Women of childbearing potential must have a negative serum pregnancy test performed
             within 7 days of the first dose of abemaciclib (see Appendix A for definition of
             childbearing potential). Female subjects of childbearing potential must use an
             approved contraceptive method (detailed in Appendix A) for the duration of the study
             and an additional 3 weeks after the final dose of abemaciclib.

         14. Subjects with hepatitis B must have an HBV viral load < 100 IU/mL by PCR during
             screening

         15. Must have adequate organ and hematopoietic function as defined below:

        Exclusion Criteria:

          -  1. Any history of a serious medical or psychiatric condition that would prevent the
             subject from signing the informed consent form 2. Pregnant or breastfeeding 3. Use of
             any chemotherapy within 3 weeks prior to the first study treatment date 4. Use of any
             experimental therapy within 4 weeks or 5 half-lives, whichever is longer, prior to the
             first study treatment date 5. Use of radiation within 2 weeks prior to the first study
             treatment date (4 weeks if radiation to liver as per section 4.1) 6. Prior treatment
             with a CDK 4/6 inhibitor 7. Prior treatment with a PD-1 or PD-L1 inhibitor 8. Those
             who have not recovered from adverse events < Grade 1 from prior therapy, with the
             exceptions of alopecia of any grade or stable peripheral neuropathy < Grade 2 9.
             Subjects may not receive concomitant anticancer agents or radiation. Antiviral agents
             aimed at treating infectious hepatitis are permitted 10. History of or suspected
             hypersensitivity to nivolumab or abemaciclib 11. Uncontrolled ascites 12. Esophageal
             varices requiring treatment within the past 6 months (banding or medication) 13.
             Subjects with uncontrolled brain metastases. Subjects with brain metastases must have
             stable neurological status following local therapy (surgery or radiation) for at least
             4 weeks prior to first study treatment and must be off of steroids related to the
             brain metastases.

             14. Any concurrent condition requiring the continued or anticipated use of systemic
             steroids beyond physiologic replacement dosing (excluding non-systemic inhaled,
             topical skin, nasal, and/or ophthalmic corticosteroids). All other systemic
             corticosteroids above physiologic replacement dosing must be discontinued at least 4
             weeks prior to first study treatment 15. Active drug or alcohol use or dependence as
             documented in the chart that, in the opinion of the investigator, would interfere with
             adherence to study requirements 16. Active bacterial or fungal infection requiring IV
             therapy at the start of protocol treatment 17. A second primary malignancy that, in
             the judgment of the investigator, may affect the interpretation of results 18. Any
             illness or condition that in the opinion of the investigator may affect the safety of
             the subject or the evaluation of any study endpoint (e.g. 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) 19. Personal history of ventricular tachycardia, ventricular
             fibrillation, or sudden cardiac arrest 20. Prior organ allograft or allogeneic bone
             marrow transplantation 21. Pre-existing thyroid abnormality with thyroid function that
             cannot be maintained in the normal range with medication 22. Active autoimmune
             disease, except for vitiligo, type 1 diabetes mellitus, asthma, atopic dermatitis, or
             endocrinopathies manageable by hormone replacement; other autoimmune conditions may be
             allowable at the discretion of the Principal Investigator 23. Any other conditions
             judged by the investigator that would limit the evaluation of the subject 24. HIV
             positive by PCR