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