The purpose of this study is to find out what effects, good and/or bad, Palbociclib (Ibrance) (formerly known as PD0332991) has on the patient and on the liposarcoma.

Palbociclib is an investigational drug. An investigational drug is a medication that has not been approved for marketing by the Food and Drug Administration (FDA). Palbociclib blocks a protein called CDK4 which is part of a pathway in liposarcoma cells that is over-active. The investigators hope that blocking CDK4 will shut down this pathway in the liposarcoma cells and stop tumors from growing. Palbociclib is an oral medication.

Inclusion Criteria:

• A diagnosis of liposarcoma confirmed at MSKCC. Because myxoid / round cell liposarcoma does not have significant CDK4 amplification, patients with this subtype are not eligible.
• Metastatic and/or locally advanced or locally recurrent disease that is not surgically resectable, with evidence of disease progression, either clinically or radiographically, as determined by the investigator
• All patients must have measurable disease as defined by RECIST 1.1. Measurable disease is defined as at least one lesion that can be accurately measured in at least one dimension (longest diameter to be recorded). Each lesion must be >10 mm when measured by CT, MRI or caliper measurement by clinical exam; or >20 mm when measured by chest x-ray. Lymph nodes must be > 15 mm in short axis when measured by CT or MRI.
• A minimum of 1 prior systemic regimen for recurrent/metastatic disease. Note: This requirement does not apply to patients enrolled in the Expansion Cohort. The last dose of systemic therapy (include targeted therapies) must have been given at least 2 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 > or = 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 ≥ 1.5x109/L Hemoglobin ≥ 9.0 g/dL WBC ≥ 3.0x109/L Platelets ≥ 100x109/L Total bilirubin ≤ 1.5 x ULN except for patients with known Gilbert syndrome AST(SGOT)/ALT(SGPT) ≤ 3 x institutional ULN Serum creatinine ≤ 1.5 x ULN or Creatinine Clearance > 50 mL/min (calculated by Cockcroft-Gault method)QTc interval ≤ 470 msec

• Patients must not have current evidence of another malignancy that requires treatment.
• The effects of Palbociclib on the developing human fetus at the recommended therapeutic dose are unknown. 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 intact Palbociclib capsules.
• Patients" tumors must express Rb, as assessed using an historical biopsy sample if available or a newly obtained tumor sample. Samples must demonstrate ≥1+ staining for Rb. Patients' tumors must also have evidence of CDK4 amplification by FISH. Note: This does not apply to patients enrolled in the Expansion Cohort.

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
• History of allergic reactions attributed to compounds of similar chemical or biologic composition to Palbociclib.
• 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.
• Patients with a history of long-QT syndrome or documented family history of long-QT syndrome. Patients who must remain on drugs that prolong the QT interval.
• Palbociclib is a substrate of CYP3A. Caution should be exercised when dosing Palbociclib concurrently with CYP3A inducers or inhibitors. Furthermore, patients who are taking concurrent medications that are strong inducers/inhibitors or substrates of CYP3A4 should be switched to alternative medications to minimize any potential risk. A list of CYP3A4 substrates, inducers and/or inhibitors is provided in Appendix B. The following medications with strong potential for interaction are not allowed: indinavir nelfinavir ritonavir clarithromycin itraconazole ketoconazole nefazodone saquinavir telithromycin carbamazepine phenobarbital phenytoin pioglitazone rifabutin rifampin St. John's wort Troglitazone

2 years