Abemaciclib in Children With DIPG or Recurrent/Refractory Solid Tumors

Inclusion Criteria for All Participants: * Patient must have measurable or evaluable disease. * Age must be ≥ 2 years and \< 25 years * Body surface area (BSA) ≥ 0.5 m\^2 * Lansky (for participants ≤ 16 years) or Karnofsky (for participants \> 16 years) performance score ≥ 40 at the time of study enrollment * Adequate organ function at the time of study enrollment as follows: * Bone marrow: Absolute neutrophil count (ANC) ≥ 1,000/μL, platelet count ≥ 75,000/μL (transfusion independent for ≥ 7 days), hemoglobin concentration ≥ 8g/dL (may be transfused) * Patients with bone marrow metastatic disease who do not meet the above criteria will be eligible to enroll in the study with the following count criteria. These patients will not be evaluable for hematologic toxicity or hematologic DLT. * ANC \> 750/μL within 7 days prior to first dose of abemaciclib * Platelet count \> 50,000/μL (may receive platelet transfusions) within 7 days prior to first dose of abemaciclib * Hemoglobin ≥ 7.5 g/dL (may receive red blood cell (RBC) transfusions) within 7 days prior to first dose of abemaciclib * Renal: Normal serum creatinine concentration based on age or glomerular filtration rate (GFR) \> 70 ml/min/1.73m\^2 * Hepatic: Total bilirubin concentration \< 1.5x the institutional upper limit of normal for age; serum glutamic pyruvic transaminase (SGPT) \< 10x the institutional upper limit of normal for patients on Stratum A. Stratum B patients must have SGPT \< 4x the institutional upper limit of normal. * Cardiac: Adequate cardiac conductivity with corrected Q-T interval (QTC) of \< 450 ms on screening ECG. * Female research participants of childbearing age must not be pregnant as confirmed by a serum or urine pregnancy test within 1 week of start of treatment. Participants must not be breast-feeding. * All patients should submit an archival tumor biopsy specimen (collected at diagnosis or relapse). Patients who have no tumor tissue available may be permitted to participate after discussion with the principal investigator. * Males or females of reproductive potential may not participate unless they have agreed to use two effective contraceptive methods. Abstinence in a non-sexually active child will be sufficient birth control. Inclusion Criteria for Stratum A (Newly Diagnosed DIPG) * Diagnosis of DIPG or high-grade glioma originating from the brainstem * Participants have had no previous treatment except corticosteroid use. Inclusion Criteria for Stratum B (Recurrent/refractory/progressive MBT (including DIPG) or ST) - Stratum B is closed to further accrual of participants * Patients must have radiologic evidence of recurrent, refractory or progressive malignant central nervous system (WHO Grade III or IV) or solid tumor. For patients with radiologic features of DIPG histologic confirmation of diagnosis is not required though biopsy is suggested if clinically indicated. * Patients with neurological deficits should have deficits that are stable for a minimum of 1 week prior to registration. * Patients who are on dexamethasone must be on a stable or decreasing dose for at least one week prior to registration. * Patients must have fully recovered from the acute toxic effects of chemotherapy, immunotherapy, or radiotherapy prior to entering this study. * Myelosuppressive chemotherapy: Patients must have received their last dose of known myelosuppressive anticancer chemotherapy at least 21 days prior to study registration or at least six weeks if nitrosourea. At least two weeks must have lapsed if patients received lower dose oral etoposide (50 mg/2) without experiencing evidence of myelosuppression (i.e. neutropenia or requiring transfusion with blood products) * Biologic agent: Patient must have recovered from any toxicity potentially related to the agent and received their last dose of the biologic agent ≥ 7 days prior to study registration. * Monoclonal antibody treatment: At least three half-lives must have elapsed prior to registration. * Radiation: Patient has received radiation therapy prior to study registration. Patients must have had their last fraction of local irradiation to the primary tumor ≥ 3 months prior to registration, their last fraction of craniospinal irradiation (\>24Gy) or total body irradiation \> 3 months prior to registration or \> 6 wks for therapeutic doses of metaiodobenzylguanidine (MIBG). Patient has not received focal irradiation for symptomatic metastatic sites within 14 days prior to registration. * Bone Marrow Transplant: Patient must be ≥ 3 months since high dose chemotherapy and peripheral blood stem cell rescue prior to registration. * Autologous stem cell transplant following myeloablative therapy within 3 months prior to the first dose of abemaciclib or prior allogeneic stem cell transplant at any time. Patients who received stem cell reinfusion following non-myeloablative therapy are eligible once they meet peripheral blood count criteria. * Growth factors: Patients must be off all colony forming growth factors(s) for at least 1 week prior to registration (filgrastim, sargramostim, erythropoietin) and at least 2 weeks for long-acting formulations (e.g. Neulasta). Exclusion Criteria: * Patients with uncontrolled infection * Patients with any concomitant significant medical illness that in the investigator's opinion cannot be adequately controlled with appropriate therapy, or that would impair the evaluation of side effects related to this treatment, alter drug metabolism or the tolerance to this treatment * Patients receiving any other anticancer or investigational drug therapy * Prior therapy with abemaciclib * Known mutation of Rb in tumor tissue * Prior history of QTC prolongation or QTC\>450 ms on screening ECG.