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).