Study of Abemaciclib in Dedifferentiated Liposarcoma

Inclusion Criteria: * A diagnosis of dedifferentiated liposarcoma confirmed at MSKCC. * Metastatic and/or locally advanced or locally recurrent disease that is not surgically resectable. * All patients must have measurable disease as defined by RECIST 1.1. Patients must also have evidence of disease progression by RECIST 1.1 within 6 months of first dose of study drug. * Any number of prior therapies (including none) is permitted. The last dose of systemic therapy (include targeted therapies) must have been given at least 2 4 weeks prior to initiation of therapy. Patients receiving BCNU or mitomycin C must have received their last dose of such therapy at least 6 weeks prior to initiation of therapy. * Patients with brain metastasis that have been treated with definitive surgery or radiation and have been clinically stable for 3 months are eligible. * Age ≥ 18 years. * ECOG performance status 0 or 1 * Adequate organ and marrow function as defined below (ULN indicates institutional upper limit of normal): * Absolute neutrophil count ≥ 1.510\^9/L * Hemoglobin ≥ 8.0 g/dL * WBC ≥ 3.0 x 10\^9/L * Platelets ≥ 100 x 10\^9/L * Total bilirubin ≤ 1.5 x ULN except for patients with known Gilbert syndrome * AST(SGOT)/ALT(SGPT) ≤ 3 x institutional ULN * Creatinine ≤ 1.5 x ULN or Creatinine Clearance \> 50 mL/min (calculated by Cockcroft-Gault method) * Patients must not have current evidence of another malignancy that requires treatment. * Women of child-bearing potential and men must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence). Women must not breast feed while on study. * Ability to understand and the willingness to sign a written informed consent document. * Ability to swallow capsules Exclusion Criteria: * Patients who have not recovered from adverse events of prior therapy to ≤ NCI CTCAEv4.0 Grade 1. * Patients receiving any other investigational agents. * Patients who have received prior treatment with a selective CDK4 inhibitor * Uncontrolled intercurrent illness including, but not limited to, known ongoing or active infection, including HIV, active hepatitis B or C, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia (specifically, atrial fibrillation or ventricular dysrhythmias except ventricular premature contractions), or psychiatric illness/social situations that would limit compliance with study requirements. * Pregnant women and women who are breast-feeding.