-  INCLUSION CRITERIA:

          -  Participants must have Kaposi sarcoma confirmed by the Laboratory of Pathology, NCI

          -  All participants should have at least five measurable cutaneous KS lesions with no
             previous local radiation, surgical or intralesional cytotoxic therapy that would
             prevent response assessment for that lesion.

          -  Measurable disease by the criteria proposed by the AIDS Clinical Trials Group Oncology
             Committee.

          -  Participants may be HIV positive or negative.

          -  Participants must be able to swallow oral medications

          -  For all groups, participants must have adequate organ and marrow function as defined
             below:

               -  Absolute neutrophil count >1,000/mcL

               -  Platelets >75,000/mcL

               -  Hemoglobin >= 8gm/dL

               -  Total bilirubin <= 1.5 upper limit of normal unless the participant is receiving
                  a protease inhibitor known to be associated with increased bilirubin (e.g.
                  atazanavir), in which case total bilirubin <= 7.5 mg/dL with direct fraction <=
                  0.7

               -  AST(SGOT)/ALT(SGPT) <3 X institutional upper limit of normal

               -  Creatinine within normal institutional limits OR

               -  Creatinine clearance >45 mL/min/1.73 m2 as estimated by either Cockroft-Gault or
                  24-hour urine collection for participants with creatinine levels above
                  institutional normal

               -  Cardiac ejection fraction > 45% by echocardiogram

          -  For phase 1: Participants must have received at least 1 prior line of systemic therapy
             for KS with either plateau in response, progressive disease, or inadequate response to
             treatment. Previous local therapy or radiation is not considered systemic therapy.

          -  For phase 2: Group 2a: Participants must have received at least 1 prior line of
             systemic therapy for KS with either plateau in response, relapsed disease, progressive
             disease, or inadequate response to treatment

          -  For phase 2: Group 2b: Participants have not received prior systemic therapy for KS.
             Previous local therapy or radiation is not considered systemic therapy.

          -  Age >18 years

          -  ECOG performance status <= 2 (Karnofsky >= 60%.

          -  Human immunodeficiency virus (HIV)-infected participants on effective anti-retroviral
             therapy are eligible for this trial.

          -  Willingness to adhere to ART

          -  For all arms of the study, participants must have received ART for 8 weeks prior to
             enrollment, with no evidence of KS improvement over the most recent 4 weeks

          -  For participants with evidence of chronic hepatitis B virus (HBV) infection, the HBV
             viral load must be undetectable on suppressive therapy, if indicated.

          -  Participants with a history of hepatitis C virus (HCV) infection must have been
             treated and cured. For participants with HCV infection who are currently on treatment,
             they are eligible if they have an undetectable HCV viral load.

          -  No uncontrolled severe concurrent bacterial, viral, or fungal infections.

          -  Participants with known history or current symptoms of cardiac disease, or history of
             treatment with cardiotoxic agents, should have a clinical risk assessment of cardiac
             function using the New York Heart Association Functional Classification. To be
             eligible for this trial, participants should be class 2B or better.

          -  The effects of abemaciclib on the developing human fetus are unknown. For this reason
             and because CDK inhibitors are known to be teratogenic, persons of child-bearing
             potential and their sexual partners must agree to use adequate pregnancy contraception
             (hormonal or barrier method of birth control; abstinence) prior to study entry and for
             the duration of study treatment. Should a person become pregnant or suspect they are
             pregnant while they or their partner is receiving study drug in this study, the
             pregnant person should inform their treating physician immediately. Participants with
             sexual partners of childbearing potential treated or enrolled on this protocol must
             also agree to use adequate contraception prior to the study, for the duration of study
             treatment, and 4 months after completion of abemaciclib administration.

          -  Ability to understand and the willingness to sign a written informed consent document.

        Exclusion Criteria:

          -  Participants who have had chemotherapy or immunotherapy within 3 weeks prior to
             entering the study.

          -  Participants who received radiotherapy must have completed and fully recovered from
             the acute effects of radiotherapy. A washout period of at least 14 days is required
             between end of radiotherapy and enrollment.

          -  Participants who have not recovered from adverse events due to prior anti-cancer
             therapy (i.e., have residual toxicities > Grade 1) with the exception of alopecia.

          -  Participants who are receiving any other investigational agents.

          -  History of severe allergic reactions attributed to compounds of similar chemical or
             biologic composition to CDK inhibitor.

          -  Participants receiving any medications or substances that are strong/moderate
             inhibitors of CYP3A4 are ineligible. Because the lists of these agents are constantly
             changing, it is important to regularly consult a frequently-updated medical reference.
             As part of the enrollment/informed consent procedures, the participant will be
             counseled on the risk of interactions with other agents, and what to do if new
             medications need to be prescribed or if the participant is considering a new
             over-the-counter medicine or herbal product.

          -  Participants with serious and/or uncontrolled severe intercurrent illness that in the
             judgement of the investigator would preclude participation in the study.

          -  No active KSHV-associated multicentric Castleman disease, KSHV-associated inflammatory
             cytokine syndrome or primary effusion lymphoma.

          -  Participants with psychiatric illness/social situations that would limit adherence
             with study requirements.

          -  Pregnant persons are excluded from this study because abemaciclib is CDK inhibitor
             with the potential for teratogenic or abortifacient effects. Because there is an
             unknown but potential risk for adverse events in nursing infants secondary to
             treatment of the nursing person with abemaciclib, breastfeeding should be discontinued
             if the nursing person is treated with abemaciclib.

          -  Participants with a prior or concurrent malignancy whose natural history or treatment
             does not have the potential to interfere with the safety or efficacy assessment of the
             regimen are eligible for this trial

          -  Participants with interstitial lung disease