Inclusion Criteria:

- Phase 1 portion:

- Participants with a cytologic or histologic diagnosis of nonsquamous Non-small
Cell Lung Cancer (NSCLC) which is classified as Stage IV according to the
seventh edition of the American Joint Committee on Cancer (AJCC) classification
and for whom the combination of pemetrexed and cisplatin is deemed to be
appropriate

- Participants with histologic or cytologic diagnosis of malignant mesothelioma
which is unresectable

- Participants with histologic or cytologic diagnoses of advanced or metastatic
solid tumors who are not candidates for any standard therapy and for whom the
combination with pemetrexed and cisplatin is deemed to be appropriate

- Phase 2 portion:

- Have a histological diagnosis of NSCLC other than predominantly squamous cell
histology which is classified as Stage IV according to the seventh edition of
the American Joint Committee on Cancer (AJCC) classification

- Be eligible for a first line of palliative treatment with a platinum doublet

- Have archived tumor tissue (not cytology)

- Phase 1 participants can have measurable or nonmeasurable disease. Phase 2
participants must have at least 1 measurable lesion according to Investigational New
Drug (IND) (Response Evaluation Criteria in Solid Tumors [RECIST]) definitions.
Tumor lesions located in a previously irradiated area can be considered measurable if
they are new or if have shown unequivocal progression.

- Have a performance status of less than or equal to one on the Eastern Cooperative
Oncology Group (ECOG) scale

- Have adequate hematologic, hepatic and renal organ function

- Prior radiation therapy for treatment of cancer is allowed to less than 25% of the
bone marrow, and participants must have recovered from the acute toxic effects of
their treatment prior to study enrollment. Prior radiation to the whole pelvis is
not allowed. Prior radiotherapy must be completed at least 4 weeks before study
entry

- For women: Must be surgically sterile, postmenopausal, or compliant with a
highly reliable contraceptive method (failure rate less than 1%) during and for 6
months after the treatment period; must have a negative serum or urine pregnancy test
within 7 days before study enrollment and must not be breast-feeding.For men: Must be
surgically sterile or compliant with a contraceptive regimen during and for 6 months
after the treatment period

Exclusion Criteria:

- Have serious preexisting medical conditions or serious concomitant systemic disorders
that would compromise the safety of the patient or his/her ability to complete the
study, at the discretion of the investigator (for example, unstable angina pectoris
or uncontrolled diabetes mellitus). Special attention should be paid to kidney and
heart conditions that may be worsened with cisplatin treatment or hydration

- Have central nervous system (CNS) metastases (unless the participant has completed
successful local therapy for CNS metastases and has been off corticosteroids for at
least 4 weeks before starting study therapy). A screening computed tomography (CT)
scan or magnetic resonance imaging (MRI) before enrollment in the absence of a
clinical suspicion of brain metastases is not required

- Have current active infection that would, in the opinion of the investigator,
compromise the patient's ability to tolerate therapy

- Have known allergy to pemetrexed, cisplatin, LY2603618, or any ingredient of
pemetrexed, cisplatin, or LY2603618

- Have clinically significant (by physical exam) third-space fluid collections; for
example, ascites or pleural effusions that cannot be controlled by drainage or other
procedures prior to study entry

- Patients taking non-steroidal anti-inflammatory drugs (NSAIDs) who cannot interrupt
the treatment appropriately according to the guidelines

- Have received a recent yellow-fever vaccination (within 28 days of enrollment) or are
receiving concurrent yellow-fever vaccination

- Phase 1 portion:

- Have received more than 2 previous lines of chemotherapy for the
advanced/metastatic disease

- Have received more than 6 cycles of therapy containing an alkylating agent