Inclusion Criteria:

          -  Participants must have histologically or cytologically confirmed disease from any
             solid tumor

          -  Participants must have measurable disease in the CNS, defined as at least one lesion
             that can be accurately measured in at least one dimension as ≥10 mm .

          -  Participants must have progressive CNS lesions, as defined by one of the following:

               -  Patients may have multiple progressive CNS lesions, some of which have been
                  treated by SRS or surgery. Patients are eligible if they have one or more
                  un-treated (by surgery or SRS) progressive lesions that is measurable.

               -  Patients have measurable residual or progressive lesions after surgery.

               -  Patients who have had prior WBRT and/or SRS are eligible but there needs to be
                  unequivocal evidence of progression of at least one lesion treated by radiation
                  (e.g. tissue diagnosis). Biopsy can be considered for definitive diagnosis.

               -  Patients who have previously been treated with systemic therapy for CNS
                  metastases are eligible.

          -  Age ≥ 18 years. The toxicity of palbociclib in children is unknown.

          -  ECOG performance status ≤2 (Karnofsky ≥60%, see Appendix A)

          -  Participants must have normal organ and marrow function as defined below:

               -  leukocytes ≥3,000/mcL

               -  absolute neutrophil count ≥1,500/mcL

               -  platelets ≥100,000/mcL

               -  hemoglobin >9g/dL

               -  total bilirubin ≤ 1.5 x institutional upper limit of normal

                  --- OR

               -  > 1.5 x institutional upper limit of normal allowed if direct bilirubin is within
                  normal range.

               -  AST(SGOT)/ALT(SGPT) ≤2.5 × institutional upper limit of normal

               -  creatinine within normal institutional limits

                  --- OR

               -  creatinine clearance ≥60 mL/min/1.73 m2 for participants with creatinine levels
                  above institutional normal.

               -  baseline QTc <480ms

          -  The effects of palbociclib 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 and for
             the duration of study participation. 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. Men treated or enrolled on this protocol must also
             agree to use adequate contraception prior to the study, for the duration of study
             participation, and 6 months after completion of palbociclib administration.

          -  Ability to understand and the willingness to sign a written informed consent document.

          -  Tissue from a prior craniotomy or biopsy for genetic sequencing (at least one FFPE
             block or 15 unstained slides). Patients previously assessed for genetic sequencing who
             meet requirements of section 9.2.1 do not need to have additional tissue available for
             prospective genetic sequencing.

          -  Presence of alteration in CDK pathway (amplifications in CDK4, CDK6, CCND1, CCND2,
             CCND3 or CCNE1 or loss of CDKN2A)

          -  Patients with progressive extracranial disease will not be excluded.

          -  Stable corticosteroids for at least 7 days

        Exclusion Criteria:

          -  Prior treatment with CDK4/6 inhibitor

          -  Participants who have had chemotherapy, immunotherapy or radiotherapy within 2 weeks
             (6 weeks for nitrosoureas or mitomycin C) prior to entering the study or those who
             have not recovered from adverse events due to agents administered more than 2 weeks
             earlier.

          -  Participants who are receiving any other investigational agents

          -  Participants who are receiving other concurrent chemotherapies or immunotherapies for
             their cancer (except for patients who will receive letrozole, anastrozole, exemestane,
             tamoxifen, fulvestrant, trastuzumab, bisphosphonates, or ovarian suppression therapy)

          -  Leptomeningeal involvement of cancer

          -  History of allergic reactions attributed to compounds of similar chemical or biologic
             composition to palbociclib (including abemaciclib)

          -  Participants receiving any medications or substances that are moderate or strong
             inhibitors or inducers of CYP3A isoenzymes are ineligible. Lists including medications
             and substances known or with the potential to interact with the CYP3A isoenzymes are
             provided in Appendix C, and can also be found within section 5.4. Because the lists of
             these agents are constantly changing, it is important to regularly consult a
             frequently-updated list such as http://medicine.iupui.edu/clinpharm/ddis/table.aspx;
             medical reference texts such as the Physicians' Desk Reference may also provide this
             information. As part of the enrollment/informed consent procedures, the patient will
             be counseled on the risk of interactions with other agents, and what to do if new
             medications need to be prescribed or if the patient is considering a new
             over-the-counter medicine or herbal product.

          -  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 the effect of palbociclib on a
             developing fetus is unknown. Because there is an unknown but potential risk for
             adverse events in nursing infants secondary to treatment of the mother with
             palbociclib, breastfeeding should be discontinued if the mother is treated with
             palbociclib.

          -  HIV-positive participants on combination antiretroviral therapy are ineligible because
             of the potential for pharmacokinetic interactions with palbociclib. In addition, 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.

          -  Current use of drugs that are known to prolong the QT interval (See Appendix C)

          -  Unable to undergo MRI scans.

          -  QTc>480 msec (based on the mean value of the triplicate ECGs), family or personal
             history of long or short QTc prolongation, or Torsade de Pointes (TdP).

          -  Uncontrolled electrolyte disorders that can compound the effects of QTc-prolonging
             drug (eg. hypocalcemia, hypokalemia, hypomagnesemia