Patient description: A Pap smear in a 54-year-old woman revealed abnormal cervical squamous intraepithelial/glandular lesion. She tested positive for HPV 16. She is sexually active with her husband and has 4 children. She is menopausal and uses no contraception. She smokes and drinks alcohol frequently. She is otherwise healthy. She was offered conization.

Clinical trial eligibility criteria: eligible ages (years): 18.0-999.0,  Carcinoma, Non-small Cell Lung For all Parts: The participant must have stage IV non-small cell lung cancer (NSCLC) For Part A (abemaciclib + pemetrexed): Non-squamous subtypes only. The participant must have received at least one but no more than three prior therapies for advanced/metastatic NSCLC For Part B (abemaciclib + gemcitabine): Any subtype. The participant must have received at least one but not more than three prior therapies for advanced/metastatic NSCLC For Part C (abemaciclib + ramucirumab): Any subtype. The participant must have received at least two but not more than three prior therapies for advanced/metastatic NSCLC For Part D (abemaciclib + LY3023414): Any subtype. The participant must have received at least two, but not more than three prior therapies for advanced/metastatic NSCLC. The participant must not have received prior treatment with any phosphoinositide 3-kinase (PI3K) or mammalian target of rapamycin (mTOR) inhibitor For Part E (abemaciclib + pembrolizumab): Any subtype. The participant must have received at least one but no more than three prior therapies for advanced/metastatic NSCLC Have either measureable or nonmeasurable disease as defined by the Response Evaluation in Solid Tumors (RECIST v1.1) Have adequate organ function including Hematologic: Absolute neutrophil count (ANC) 1.5 x 109/liter (L), platelets 100 x 109/L, and hemoglobin 8 gram/deciliter (g/dL) Hepatic: Bilirubin 1.5 times upper limits of normal (ULN), alanine aminotransferase (ALT) and aspartate transaminase (AST) 3.0 times ULN. For participants with tumor involvement of the liver, AST and ALT equaling ≤5.0 times ULN are acceptable. Alkaline phosphatase ≤5.0 times ULN for participants with tumor involvement of the bone is acceptable  Have a personal history of any of the following conditions: presyncope or syncope of either unexplained or cardiovascular etiology, ventricular arrhythmia (including but not limited to ventricular tachycardia and ventricular fibrillation), or sudden cardiac arrest. Exception: Participants with controlled atrial fibrillation for >30 days prior to study treatment are eligible Parts A, B, D and E: Have central nervous system (CNS) metastasis with development of associated neurological changes 14 days prior to receiving study drug Have a history of any other cancer (except non-melanoma skin cancer or carcinoma in-situ of the cervix or breast), unless in complete remission with no therapy for a minimum of 3 years Is pregnant or breastfeeding, or expecting to conceive or father children within the projected duration of the trial, starting with the pre-screening or screening visit through 3 to 4 months after the last dose of trial treatment (as appropriate) Have active bacterial, fungal, and/or known viral infection (for example, human immunodeficiency virus [HIV] antibodies, hepatitis B surface antigen [HBSAg], or hepatitis C antibodies). Screening is not required for enrollment Parts A, B, C, and E: Have QTc interval of > 470 millisecond (msec) on screening electrocardiogram (ECG). Part D participants have QTc interval of >450msec on screening ECG. Additional For Part C History or evidence of cardiovascular risk including any of the following History of acute coronary syndromes (including myocardial infarction and angina), coronary angioplasty, or stenting within 6 months prior to enrollment History or evidence of current ≥Class II congestive heart failure as defined by New York Heart Association