A Study of LY2835219 (Abemaciclib) in Combination With Therapies for Breast Cancer That Has Spread

Inclusion Criteria:

* Have a diagnosis of hormone receptor positive (HR+), human epidermal growth factor receptor 2 (HER2) negative metastatic breast cancer for Parts A to E, G, and I.
* Have a diagnosis of human epidermal growth factor receptor 2 (HER2) positive metastatic breast cancer for Parts F and H.
* For Part A (LY2835219 + letrozole): Except for ongoing therapy with letrozole, the participant must not have received prior systemic endocrine therapy for metastatic disease.
* For Part B (LY2835219 + anastrozole): Except for ongoing therapy with anastrozole, the participant must not have received prior systemic endocrine therapy for metastatic disease.
* For Part C (LY2835219 + tamoxifen): The participant may have received prior systemic endocrine therapy for metastatic disease and may be receiving ongoing therapy with tamoxifen.
* For Part D (LY2835219 + exemestane): The participant must have received prior systemic endocrine therapy with at least one nonsteroidal aromatase inhibitor (anastrozole, letrozole) for metastatic disease and may be receiving ongoing therapy with exemestane.
* For Part E (LY2835219 + exemestane + everolimus): The participant must have received prior systemic endocrine therapy with at least one nonsteroidal aromatase inhibitor (anastrozole, letrozole) for metastatic disease and may be receiving ongoing therapy with either exemestane or exemestane + everolimus.
* For Part F (LY2835219 + trastuzumab):The participant must have received at least 1 chemotherapy regimen for metastatic disease and may be receiving ongoing therapy with trastuzumab. The participant must have an estimated left ventricular ejection fraction within the normal range by either echocardiogram or multigated acquisition (MUGA) scan
* For Part G (abemaciclib + LY3023414 + fulvestrant): The participant may have received prior systemic endocrine therapy with at least one nonsteroidal aromatase inhibitor (anastrozole, letrozole) for metastatic disease.
* For Part H: (abemaciclib + trastuzumab + pertuzumab): The participant must have received at least 1 chemotherapy regimen for metastatic disease. The participant may be receiving ongoing therapy with trastuzumab and/or pertuzumab at the time of study entry. The participant must have an estimated left ventricular ejection fraction (LVEF) within the normal range by either echocardiogram or multigated acquisition (MUGA) scan.
* For Part I (abemaciclib + endocrine therapy): The participant must have demonstrated evidence of disease progression on a Cyclin Dependent Kinase 4 (CDK4) and Cyclin Dependent Kinase 6 (CDK6) inhibitor (either palbociclib or ribociclib) plus endocrine therapy for advanced or metastatic disease as the most recent therapy immediately preceding study entry. The participant should remain on the current endocrine therapy while receiving abemaciclib.
* For Parts A, B, C, D, E, and F: Have either measureable disease or nonmeasureable but evaluable bone disease as defined by the Response Evaluation Criteria in Solid Tumors (RECIST 1.1)
* For Part G, H, and I: Have measureable disease as defined by RECIST 1.1.
* For all Parts except Part F and H: Participants must have either post-menopausal status or pre-menopausal status if continuing or beginning ovarian suppression with a luteinizing hormone-releasing hormone (LHRH) agonist such as goserelin.
* Parts H, and I: Must be able and willing to undergo mandatory tumor biopsies prior to study treatment and at the time of discontinuation from study treatment.
* Have adequate organ function, including:

  * Hematologic: Absolute neutrophil count (ANC) ≥1.5 x 10\^9/liter (L), platelets ≥ 100 x 10\^9/L, and hemoglobin ≥ 8 gram/deciliter (g/dL).
  * Hepatic: Bilirubin ≤1.5 times upper limits of normal (ULN), alanine aminotransferase (ALT) ≤ 3.0 times ULN.
  * Renal: Serum creatinine ≤ 1.5 times ULN.
* Have a performance status of ≤1 on the Eastern Cooperative Oncology Group (ECOG) scale.
* Have discontinued all previous therapies for breast cancer (including chemotherapy, radiotherapy, immunotherapy, and investigational therapy), except for ongoing corresponding combination therapy, for at least 21 days for myelosuppressive agents or 14 days for nonmyelosuppressive agents prior to receiving study drug(s), and recovered from the acute effects of therapy (until the toxicity resolves to either baseline or at least Grade 1) except for residual alopecia or peripheral neuropathy. For Part F and H: concurrent treatment with trastuzumab emtansine (T-DM1) is not allowed.

Exclusion Criteria:

* Have metastatic breast cancer with severe organ dysfunction as assessed by symptoms and signs, laboratory studies, and rapid progression of the disease.
* Have brain metastasis without prior radiotherapy.
* For Parts A, B, C, D, E, G and I: Have received prior systemic chemotherapy for metastatic disease. However, the participant may have received prior systemic chemotherapy in the neoadjuvant or adjuvant setting.
* For Parts A, B, C, D, E, F, H: Have received prior therapy with a CDK4/6 inhibitor, Part G: Have received prior therapy with fulvestrant or any PI3K and/or mTOR inhibitor (including LY3023414); Part I: Have received prior treatment with abemaciclib in any setting.
* Have serious preexisting medical conditions that, in the judgment of the investigator, would preclude participation in this study (including interstitial lung disease (ILD), severe dyspnea at rest or requiring oxygen therapy or, history of major surgical resection involving the stomach or small bowel).
* Have central nervous system (CNS) metastasis with either radiotherapy or development of neurological changes ≤14 days prior to receiving study treatment. Participants may be receiving a stable dose of corticosteroids. Screening of asymptomatic participants without history of CNS metastasis is not required. Untreated CNS metastases are not permitted.
* For Parts F and H: Cardiac disease including myocardial infarction within 6 months, unstable angina, or New York Heart Association (NYHA) Grade II or greater functional impairment.
* For Part G: Have type 1 diabetes mellitus or a history of gestational diabetes mellitus. Participants with a type 2 diabetes mellitus are eligible if adequate control of blood glucose level is obtained with oral therapy as documented by Hemoglobin A1c \<7%.
* For Part G: Have a baseline electrocardiogram (obtained from Day -14 to Day -1) with any of the following abnormal findings: ventricular arrhythmia, evidence of acute myocardial ischemia, heart block (of any degree), or QTc prolongation (defined as QTcB ≥450 milliseconds).