The purpose of this study is to find a recommended dose level and schedule of dosing LY3023414 that can safely be taken by participants with advanced or metastatic cancer. The study will also explore the changes to various markers in blood cells and potentially tumor cells. Finally, the study will help document any antitumor activity this drug may have.

In Part A of this study, participants with advanced/metastatic cancer (including lymphoma) will receive increasing doses of LY3023414. In Part B, LY3023414 will be explored in different types of cancer, including breast and lung cancer, lymphoma and mesothelioma.

Inclusion Criteria:

• Parts A, A2 & B1: Participants must have pathological evidence of a diagnosis of advanced and/or metastatic cancer and must be, in the judgment of the investigator, an appropriate candidate for experimental therapy
• Part B2: Participants must have advanced, recurrent, or metastatic breast cancer that is refractory to aromatase inhibitors (AI) with either disease recurrence or disease progression; must be hormone receptor positive (HR+) and human epidermal growth factor receptor 2 (HER2)-negative; must be of postmenopausal status or beginning ovarian suppression with a luteinizing hormone-releasing hormone (LHRH) agonist
• Part B3 only: Participants must have malignant pleural or peritoneal mesothelioma
• Part B4 only: Participants must have malignant pleural or peritoneal mesothelioma and appropriate candidate for treatment with cisplatin/pemetrexed; no prior systemic chemotherapy
• Part B5 only: Participants must have histologically confirmed diagnosis of B-cell iNHL, with histological subtype; prior treatment with ≥2 prior chemotherapy- or immunotherapy-based regimens for iNHL
• Part B6 only: Participants must have squamous NSCLC; documented evidence of an activating molecular aberration of the PI3K/mTOR pathway
• Parts B2, B3 & B6 only: Must have adequate tumor tissue sample from archival biopsy available, or willingness to undergo a fresh tumor biopsy
• Parts B3, B4, B5 & B6: No previous treatment with any PI3K and/or mTOR inhibitor
• Part B7: Must have a diagnosis of HR+ and HER2- breast cancer; have locoregionally recurrent disease not amenable to resection or radiation therapy with curative intent or metastatic disease; no previous treatment or currently receiving 1 of the following treatments for locoregionally recurrent or metastatic breast cancer (chemotherapy, endocrine therapy, CDK4/6 inhibitor, and PI3K and/or mTOR inhibitor)
• Measurable or nonmeasurable disease as defined by the Response Evaluation Criteria in Solid Tumors (RECIST Version 1.1), modified RECIST or Revised Response Criteria for Malignant Lymphoma
• Have adequate organ function, including: Absolute neutrophil count (ANC) at least 1.5 x 109/Liter (L), platelets at least 100 x 109/L, and hemoglobin at least 8 grams/deciliter (g/dL); bilirubin no more than 1.5 times upper limits of normal; alanine aminotransferase (ALT) and aspartate aminotransferase (AST) no more than 2.0 times upper limits of normal; Serum creatinine no more than 1.5 times upper limits of normal or calculated creatinine clearance >45 milliliters/minute (mL/min)
• Have a performance status of at least 1 on the Eastern Cooperative Oncology Group (ECOG) scale and life expectancy >6 months
• Have discontinued all previous cancer therapies (except nonsteroidal aromatase inhibitors for participants in Part B2), and any agents that have not received regulatory approval for any indication, for at least 21 days or 5 half lives prior to study enrollment, whichever is shorter, and recovered from the acute effects of therapy. Participants must have discontinued mitomycin-C or nitrosourea therapy for at least 42 days
• Are able to swallow capsules

Exclusion Criteria:

• Have serious preexisting medical conditions
• Have symptomatic central nervous system (CNS) malignancy (with the exception of medulloblastoma) or metastasis (screening not required).
• Have known acute or chronic leukemia or current hematologic malignancies (except iNHL for patients in Part B5) that, in the judgment of the investigator and sponsor, may affect the interpretation of results
• Have an active fungal, bacterial, and/or known viral infection
• Have a second primary malignancy that in the judgment of the investigator and sponsor may affect the interpretation of results (Part B only)
• Part B1 only: No concomitant medications that are strong inhibitors or inducers of cytochrome P450 3A4 (CYP3A4) or midazolam
• Intolerance to any previous treatment with any phosphatidylinositol-3-kinase (PI3K) and/or mammalian target of rapamycin (mTOR) inhibitor.
• Participants with active alcohol abuse, as determined by the investigator
• Have a history of New York Heart Association (NYHA) Class ≥3, unstable angina, or myocardial infarction (MI) in 6 months prior to study drug administration
• Have QT corrected interval of >450 milliseconds (msec) on screening electrocardiogram (ECG)
• Have insulin-dependent diabetes mellitus or a history of gestational diabetes mellitus.
• Part B only: Hypersensitivity to study drugs given in combination with LY3023414