Study of the CDK4/6 Inhibitor Abemaciclib in Solid Tumors Harboring Genetic Alterations in Genes Encoding D-Type Cyclins or Amplification of CDK4 or CDK6

Inclusion Criteria: * Participants must have a histologically or cytologically confirmed advanced solid tumor of a non-breast origin, for which standard therapy proven to provide clinical benefit does not exist or is no longer effective. * For enrollment to Arm 1: Participants must have a confirmed CCND1, 2, or 3 high-level amplification, CCND1 mutation, or a CCND1 splice variant expected to lead to nuclear retention of cyclin D1 protein, via DFCI/BWH OncoPanel or any CLIA-certified method. * For enrollment to Arm 2: Participants must have a confirmed CDK4 or CDK6 high-level amplification, identified via DFCI/BWH OncoPanel or any CLIA-certified method. * Participants must have evaluable or measurable disease. * Age ≥ 18 years. * ECOG performance status of 0-1 (see APPENDIX A). * Participants must have normal organ and marrow function as defined below: * Absolute neutrophil count ≥1,500/mcL * Platelets ≥100,000/mcL * Total bilirubin ≤ 1.5 × institutional upper limit of normal (ULN) * AST(SGOT)/ALT(SGPT) ≤ 2.5 × institutional (ULN) -OR- * AST(SGOT)/ALT(SGPT) ≤ 5 × institutional (ULN) if liver metastases are present * Serum Creatinine ≤ 1.5 × institutional ULN -OR- * Creatinine clearance ≥ 60 mL/min (Cockroft-Gault Equation) * The effects of abemaciclib on the developing human fetus are unknown. For this reason women of child-bearing potential and men must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) prior to study entry, for the duration of study participation, and for 3 months after completion of abemaciclib administration. Should a woman become pregnant or suspect she is pregnant while she or her partner is participating in this study, she should inform her treating physician immediately. A negative serum pregnancy test is required for women of childbearing potential prior to study entry. * Ability to understand and the willingness to sign a written informed consent document. * Ability to swallow and retain oral medication. Exclusion Criteria: * Participants who have had chemotherapy, biologic therapy, investigational agents, radiotherapy, or major surgery within 4 weeks (6 weeks for nitrosoureas or mitomycin C) prior to entering the study. * Participants who have had oral targeted therapy or oral tyrosine kinase inhibitors (TKIs) within 5 half-lives prior to entering the study. * Participants who have received prior treatment with a CDK4/6 inhibitor. * Participants must have recovered to eligibility levels from prior toxicity or adverse events as a result of previous treatment prior to entering the study. * Participants who are receiving any other investigational agents. * Participants with hematologic lymphoma. * Participants with symptomatic CNS metastases who are neurologically unstable and/or require radiation therapy are excluded. * Participants with brain metastases that do not meet the above criteria in the opinion of the treating investigator are allowed. * Symptomatic disease is allowed as long as symptoms are controlled and stable. * History of allergic reactions attributed to compounds of similar chemical or biologic composition to abemaciclib. * Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, or psychiatric illness/social situations that would limit compliance with study requirements. * Pregnant women are excluded from this study because abemaciclib is an agent 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 mother with abemaciclib, breastfeeding should be discontinued if the mother is treated with abemaciclib. A negative serum pregnancy test is required for women of childbearing potential prior to study entry. * Participants with known HIV-positive status are ineligible because these participants are at increased risk of lethal infections when treated with marrow-suppressive therapy. Appropriate studies will be undertaken in participants receiving combination antiretroviral therapy when indicated. * Participants with known active Hepatitis B or Hepatitis C. * Participants receiving an enzyme-inducing antiepileptic drug (EIAED) who cannot be transferred to a non-EIAED (e.g., levetiracetam, lacosamide, lamotrigine, etc.) prior to the initiation of protocol therapy.