A Study of Abemaciclib (LY2835219) in Participants With Non-Small Cell Lung Cancer or Breast Cancer

Inclusion Criteria: * Have a Stage IV diagnosis of 1 of the following: Part A: NSCLC (Kirsten rat sarcoma mutant \[KRAS mt\], PD-L1+); Part B: NSCLC (squamous histology); Part C: metastatic breast cancer (HR+, HER2-); or Part D: locally advanced or metastatic breast cancer (HR+, HER2-) * Part A: must be chemotherapy naïve for metastatic NSCLC * Part B: must have received at least 1 prior therapy containing platinum-based chemotherapy for advanced/metastatic NSCLC * Part C: must have previously received prior treatment with at least 1 but no more than 2 chemotherapy regimens in the metastatic setting * Part D: cannot have received endocrine therapy or chemotherapy as treatment in the locoregionally recurrent or metastatic breast cancer disease setting. Note: Participants may be enrolled if they received prior (neo)adjuvant chemotherapy or endocrine therapy for localized disease. * Are amenable to provide tumor tissue prior to treatment and provide tumor tissue after treatment initiation (both mandatory). * Have presence of measurable disease as defined by Response Evaluation Criteria in Solid Tumors (RECIST 1.1). * Have a performance status (PS) ≤1 on the Eastern Cooperative Oncology Group (ECOG) scale. * Have discontinued all previous treatments for cancer and recovered from the acute effects of therapy. * Have an estimated life expectancy of ≥12 weeks. * For Part D: Have postmenopausal status due to surgical/natural menopause or chemical ovarian suppression (initiated 28 days prior to Day 1 of Cycle 1) with a gonadotropin-releasing hormone (GnRH) agonist such as goserelin or radiation-induced ovarian suppression. Exclusion Criteria: * Have a personal history of any of the following conditions: syncope of either unexplained or cardiovascular etiology, ventricular arrhythmia (including but not limited to ventricular tachycardia and ventricular fibrillation), or sudden cardiac arrest. Exception: subjects with controlled atrial fibrillation for \>30 days prior to study treatment are eligible. * Have central nervous system (CNS) metastasis with development of associated neurological changes 14 days prior to receiving study drug. * Have corrected QT interval of \>470 milliseconds on screening electrocardiogram (ECG). * Have history of interstitial lung disease or pneumonitis. * Have history of or active autoimmune disease, or other syndrome that requires systemic steroids or autoimmune agents for the past 2 years. * Have received a live vaccination within 30 days of study start. * Have received prior treatment with an anti PD-1, anti-programmed death ligand 1 (PD-L1), or anti cytotoxic T-lymphocyte-associated protein 4 (CTLA-4) agent. * For Part D Only: * Have initiated bisphosphonates or approved RANK ligand (RANK-L) targeted agents (for example, denosumab) \<7 days prior to Cycle 1 Day 1. * Are currently receiving or have previously received endocrine therapy for locoregionally recurrent or metastatic breast cancer. Note: A participant may be enrolled if she received prior (neo)adjuvant endocrine therapy (including, but not limited to anti-estrogens or aromatase inhibitors) for localized disease. * Are currently receiving or have previously received chemotherapy for locoregionally recurrent or metastatic breast cancer. Note: Participants may be enrolled if they received prior (neo)adjuvant chemotherapy for localized disease.