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

• Measurable disease as follows:

‣ All participants in Groups 1, 2a, or 2b should have at least five measurable cutaneous KS lesions per AIDS Clinical Trials Group Oncology Committee (ACTG) criteria 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 in Group 3 must have Stage T1 KS with at least five measurable cutaneous KS lesions per ACTG criteria and/or evaluable disease per RECIST criteria.

• 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/ALT \<3 X institutional upper limit of normal

⁃ Creatinine within normal institutional limits OR

⁃ Creatinine clearance \>45 mL/min/1.73 m\^2 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

• Prior Treatment as follows:

‣ For Phase I: 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 II: Group 2a: Individuals 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 II: Group 2b: Individuals have not received prior systemic therapy for KS. Previous local therapy or radiation is not considered systemic therapy.

• For Phase II: Group 3: Evidence of Stage T1 KS with either a) edema or ulcerated KS and/or b) extensive oral KS and/or c) visceral KS involvement

• Age \>18 years

• Eastern Cooperative Oncology Group (ECOG) performance status (PS) \<= 2 (Karnofsky \>= 60%.

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

• Willingness to adhere to ART

• All 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. Participants with HCV infection who are currently on treatment 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.

• Contraception requirements as follows:

‣ Participants of child-bearing potential (IOCBP) must agree to use a highly effective method of contraception (e.g., intrauterine device \[IUD\], hormonal, surgical sterilization, abstinence) prior to study entry, for the duration of study participation, and for up to 4 months after completion of abemaciclib administration

⁃ Participants able to father a child must with partners of childbearing potential agree to use an effective method of contraception (barrier, surgical sterilization, abstinence) for the duration of the study treatment for up to 4 months after completion of abemaciclib

∙ administration. Individuals with partners of childbearing potential should ask their partners to be on an effective birth control (hormonal, IUD, surgical sterilization).

• Because there is an unknown but potential risk for adverse events in nursing infants secondary to treatment of the nursing person with abemaciclib, nursing should be discontinued if the nursing person is treated with abemaciclib.

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