Inclusion Criteria:  Age ≥ 18 years Histologically confirmed diagnosis of dedifferentiated liposarcoma which is locally recurrent and/or metastatic. This study will accept the diagnosis made at the investigator's center. ECOG Performance Status of 0 or 1. At least one site of measurable disease on CT/MRI scan as defined by RECIST 1.1 criteria. Baseline imaging must be performed within 28 days of Day 1 of study. Adequate organ function The patient can swallow oral medications. Patients 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 randomization. Patients who received chemotherapy must have recovered (Common Terminology Criteria for Adverse Events [CTCAE] Grade ≤1) from the acute effects of chemotherapy except for residual alopecia or Grade 2 peripheral neuropathy prior to randomization. A washout period of at least 21 days is required between last chemotherapy dose and randomization (provided the patient did not receive radiotherapy). Written, voluntary informed consent Fertile men and women of childbearing potential must agree to use a highly effective method of birth control during the treatment period and for 3 weeks after last study drug administration in both sexes. Women of childbearing potential include pre-menopausal women and women within the first 2 years of the onset of menopause. Women of childbearing potential must have a negative pregnancy test ≤ seven days of the first dose of abemaciclib. Highly effective methods of birth control include an intrauterine device [IUD] or barrier method. If condoms are used as a barrier method, a spermicidal agent should be added as a double barrier protection. Measurable disease and evidence of progression of disease as defined by RECIST 1.1 (including newly diagnosed disease, new disease sites in a patient who was previously NED, or a 20% growth of existing lesions within 6 months of registration). Patients with central nervous system disease are eligible for enrollment if they have received prior radiotherapy or surgery within 3 months from completion of therapy to sites of CNS metastatic disease and are without evidence of clinical progression.  Exclusion Criteria:  Patients with documentation of well-differentiated liposarcoma only are specifically excluded, owing to its characteristically slow growth. If high grade areas are suspected (dedifferentiation), but not proved by pathology analysis (e.g. after primary resection of a well-differentiated liposarcoma), a biopsy must be performed to demonstrate the high-grade dedifferentiated disease. If there is a question regarding the diagnosis, the PI should be consulted. Patients with bulky disease who urgently need cytotoxic chemotherapy (likely with doxorubicin + ifosfamide) will be excluded from this study. This is determined by the treating physician. If there is a question regarding the appropriateness of the patient for enrollment, the PI should be consulted. Prior systemic therapy with abemaciclib or any other selective CDK4 inhibitor (such as palbociclib) Concurrent, clinically significant, active malignancies within 12 months of study enrollment Patients with severe and/or uncontrolled concurrent medical disease that in the opinion of the investigator could cause unacceptable safety risks or compromise compliance with the protocol Major surgery within 3 weeks prior to Day 1 of study or who have not recovered adequately from prior surgery Patients with resectable for curative intent disease Patients that have GI absorption disorders that would impact the administration of oral abemaciclib. Women who are pregnant or nursing/breastfeeding. Known hypersensitivity to abemaciclib. Patients with untreated central nervous system disease. Inability to comply with protocol required procedures Patients currently taking the following drugs may interact with abemaciclib. Please refer to Section 5.2 of protocol. The patient has serious and/or uncontrolled preexisting medical condition(s) that, in the judgment of the investigator, would preclude participation in this study (for example, interstitial lung disease, severe dyspnea at rest or requiring oxygen therapy, severe renal impairment [e.g. estimated creatinine clearance <30ml/min], history of major surgical resection involving the stomach or small bowel, or preexisting Crohn's disease or ulcerative colitis or a preexisting chronic condition resulting in baseline Grade 2 or higher diarrhea). The patient has active systemic bacterial infection (requiring intravenous [IV] antibiotics at time of initiating study treatment), fungal infection, or detectable viral infection (such as known human immunodeficiency virus positivity or with known active hepatitis B or C [for example, hepatitis B surface antigen positive]. Screening is not required for enrollment. The patient has a personal history of any of the following conditions: syncope of cardiovascular etiology, ventricular arrhythmia of pathological origin (including, but not limited to, ventricular tachycardia and ventricular fibrillation), or sudden cardiac arrest.