Cyclin-Dependent Kinase (CDK)4/6 Inhibitor Abemaciclib for Neurofibromatosis Type I (NF1) Related Atypical Neurofibromas

* INCLUSION CRITERIA: * All Participants * Participants must have a clinical diagnosis of NF1, i.e., participants must have at least two of the diagnostic criteria for NF1 or a confirmed NF1 mutation from a CLIA-certified laboratory: * Six or more caf(SqrRoot)(Copyright)-au-lait macules (\>= 0.5cm in prepubertal participants or \>= 1.5 cm in post pubertal participants) * Freckling in axilla or groin * A neurofibroma or plexiform neurofibroma * Optic glioma * Two or more Lisch nodules * A distinctive bony lesion (dysplasia of the sphenoid bone or dysplasia or thinning of long bone cortex) * A first-degree relative with NF1 * Measurable disease: Participants must have at least one measurable ANF defined as a lesion of at least 3 centimeters (cm) measured in one dimension. Measurability and suitability for volumetric MRI analysis of the target ANF must be confirmed with the NCI POB prior to enrolling a participant. The target ANF will be defined as the clinically most relevant ANF, which has to be amenable to volumetric MRI analysis. * Prior Therapies: * Since there is no standard effective chemotherapy for patients with NF1 and ANF, participants may be treated on this trial without having received prior medical therapy directed at their ANF. * Investigational agents/biologic therapies (not chemotherapy): Participants who have received previous investigational agents or biologic therapies are eligible for enrollment. At least 30 days or 5 half-lives must have elapsed since receiving medical therapy directed at any NF1 related tumor. Participants who received prior medical therapy for a NF1 related tumor manifestation must have recovered from the acute toxic effects of all prior therapy to grade 1 CTCAEv5 except for residual alopecia or Grade 2 peripheral neuropathy prior to enrollment before entering this study. * Chemotherapy agents: Participants who received chemotherapy must have recovered (CTCAE Grade \<=1) from the acute effects of chemotherapy except for residual alopecia or Grade 2 peripheral neuropathy prior to enrollment. A washout period of at least 28 days is required between last chemotherapy dose and enrollment (provided the participant did not receive radiotherapy). * Participants who received adjuvant radiotherapy must have completed and fully recovered from the acute effects of radiotherapy. Participants who have received previous radiation therapy are eligible for enrollment. At least 6 weeks must have elapsed since the last radiation therapy and start of treatment. The only target ANF cannot have received radiation previously. * At least 28 days must have elapsed since any major surgeries, with evidence of good wound healing. Minimally invasive biopsies and central line placements are not considered major surgeries. * Participants who have received prior treatment with abemaciclib or another specific CDK4/6 inhibitor are not eligible for enrollment * Adequate performance scale (Lansky/Karnofsky \>=70%). * Adequate organ function as defined below: --Hematologic Function: Participants must have an absolute neutrophil count \>=1500/microliter, hemoglobin \>=9 g/dL (transfusion independent, defined as not receiving blood transfusion unless related to trauma or surgeries), and platelets \>=100,000/microliter (transfusion independent, defined as not receiving platelet transfusions unless related to trauma or surgeries) * Hepatic Function: Participants must have bilirubin within 1.5 x the upper limit of normal for age, with the exception of those with Gilbert syndrome, and AST/ALT within \<= 3 x upper limit of normal. * Renal Function: Participants must have a creatinine clearance or radioisotope GFR \>=60ml/min/1.73 m2 or a normal serum creatinine based on age, described in the table below. * Age (years) is \>=12 and \<=15 then Maximum Serum Creatinine (mg/dL) = 1.2 * Age (years) \>15 Maximum Serum Creatinine (mg/dL) = 1.5 * Cardiac Function: Normal ejection fraction (ECHO or cardiac MRI) \>= 53% (or the institutional normal; if a range is given then the upper value of the range will be used); QTC or QTcF \<= 450 msec. * Willingness to avoid grapefruit or grapefruit juice during abemaciclib administration * Informed Consent: Ability of participant or Legally Authorized Representative (LAR) to understand and the willingness to sign a written informed consent document. All participants or their legal guardians (if the participant is \< 18 years old) must sign an IRB-approved document of informed consent to demonstrate their understanding of the investigational nature and the risks of this study before any protocol-related studies are performed. When appropriate, pediatric participants will be included in all discussions. * Based on animal studies, the effects of abemaciclib can cause fetal harm. For these reasons, women of child-bearing potential and men must agree to use a highly effective contraceptive method during treatment and for at least 4 months after the last dose of abemaciclib. 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 4 months after completion of abemaciclib administration * Woman participants of childbearing potential (WOCBP) must have a negative serum pregnancy test within 7 days of the first dose of abemaciclib. * A woman is considered to be of childbearing potential if she is postmenarcheal, has not reached a postmenopausal state (\>=12 continuous months of amenorrhea with no identified cause other than menopause), and has not undergone surgical sterilization (removal of ovaries and/or uterus). * Contraceptive methods may include intrauterine devices (IUD), or barrier method, a spermicidal agent should be added as a double barrier protection; abstinence is also acceptable. * Cases of pregnancy that occur during maternal exposures to abemaciclib should be reported. If a participant or spouse/partner is determined to be pregnant following abemaciclib initiation she must discontinue treatment immediately. Data on fetal outcomes and breastfeeding are to be collected for regulatory reporting and drug safety evaluation. * Phase 0 Participants * Presence of \>= 1 measurable ANF (biopsy confirmed) for which surgical removal would not likely cause significant morbidity and is clinically indicated NOTES: Definition of ANF. In addition, there will not be a requirement for confirmed CDKN2A/B deletion for study eligibility due to known biopsy sampling error and tumor heterogeneity. * Age \>= 18 years old * Phase I/II Participants * Presence of \>= 1 measurable ANF (biopsy confirmed) for whom surgical removal could cause significant clinical morbidity OR for which participant is unwilling to undergo surgical resection OR the presence of more than one distinct nodular lesion (DNL) including at least 1 biopsy proven ANF NOTES: Definition of ANF. In addition, there will not be a requirement for confirmed CDKN2A/B deletion for study eligibility due to known biopsy sampling error and tumor heterogeneity. * Age \>= 12 years old (no maximum age) with associated age-related requirements as follows: * Age \>= 12 years old with BSA \>= 0.71 m2 and able to swallow whole tablets * Willingness of participants \>= 12 years old and \<18 years old to undergo pretreatment percutaneous biopsy of ANF if deemed feasible with minimal morbidity * Willingness of participants \>= 18 years old to undergo pre-treatment and on-treatment percutaneous biopsy of ANF if deemed feasible with minimal morbidity NOTE: For participants of all ages with ANF that cannot be safely biopsied with minimal morbidity, biopsy requirement to be performed at NIH Clinical Center will be waived from eligibility criteria. In this case, review of available archival tissue by NIH Pathology will be necessary to confirm diagnosis of ANF, which is mandatory for eligibility. EXCLUSION CRITERIA: * Pregnant women, or women who intend to become pregnant during the study, are excluded from this study because of the teratogenic effects of abemaciclib. Because there is an unknown but potential risk for adverse events in nursing infants secondary to treatment of the mother with these agents, breastfeeding should be discontinued if the mother is treated on study. * May not have a NF1-related tumor such as optic pathway glioma or malignant peripheral nerve sheath tumor, which requires treatment with chemotherapy or surgery. * Serious preexisting medical condition(s) that would preclude participation in this study (for example, interstitial lung disease, severe dyspnea at rest requiring oxygen therapy, history of major surgical resection involving the stomach or small bowel that would preclude adequate absorption, or preexisting Crohn s disease or ulcerative colitis or a preexisting chronic condition resulting in baseline Grade 2 or higher diarrhea). * Uncontrolled intercurrent illness including, but not limited to, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, active bleeding diatheses or renal transplant, or psychiatric illness/social situations that would limit compliance with study requirements. * 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. * Active 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 \[for example, hepatitis B surface antigen positive\]. Participants with HIV who have adequate CD4 counts and who have no requirement for antiviral therapy will be eligible. NOTE:Screening is not required for enrollment. * Participants with interstitial lung disease * Requires treatment with strong CYP3A inhibitors or inducers * Inability to swallow tablets, since tablets cannot be crushed or broken. * Inability to undergo MRI and/or contraindication for MRI examinations following the MRI protocol. Prosthesis or orthopedic or dental braces that would interfere with volumetric analysis of target ANF on MRI. * Refractory nausea and vomiting that would limit drug administration in the opinion of the Principal Investigator * Known severe hypersensitivity to abemaciclib or any excipient of abemaciclib or history of allergic reactions attributed to compounds of similar chemical or biologic composition to abemaciclib * Clinical judgment by the investigator that the participant should not participate in the study