Abemaciclib in Patients With HIV-associated and HIV-negative Kaposi Sarcoma

* INCLUSION CRITERIA: * Participants must have Kaposi sarcoma confirmed by the Laboratory of Pathology, NCI * Measurable disease as follows: * All participants in Groups 1, 2a, or 2b or 4 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 and/or evaluable disease per RECIST criteria. * 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: * 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. * 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 * Group 2b: Individuals have not received prior systemic therapy for KS. Previous local therapy or radiation is not considered systemic therapy. * 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 * Group 4: Participants that received prior treatment with abemaciclib with evidence of response (at least a partial response per ACTG criteria and/or RECIST) to abemaciclib. * Participants in Cohort 4 must: * have been previously enrolled in Cohorts 1-3 * have at least prior documentation of PR in prior cohort * not have come off treatment for any \>Grade 3 abemaciclib-related AE. * 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 antiretroviral therapy (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 VL 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 VL. * 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. 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 or neuropathy. * 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 ART or other medications that are strong/moderate inhibitors of CYP3A4 must have dose reductions per Strong/Moderate CYP3A4 inhibitors. * 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. * Psychiatric illness/social situations that would limit adherence with study requirements. * Pregnancy * 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