This is a first-in-human, non-randomized (individuals will not be assigned by chance to study treatments), open-label (individuals will know the identity of study treatments), multicenter (more than 1 hospital work on a study), Phase 1 study. The study consists of 4 parts. Part 1 is the dose-escalation phase, which will be guided by pharmacokinetics, pharmacodynamics and safety. In part 1, safe and biologically active Phase 2 doses (recommended Phase 2 doses [RP2D]) for JNJ-42756493 will be primarily assessed. Participants will be enrolled in sequential cohorts (first cohort will receive the starting dose and subsequent cohorts will receive increased doses of JNJ-42756493). Part 2 is the Dose Confirmation Phase, which consists of a pre and post treatment tumor biopsy cohorts to confirm the RP2D based on the pharmacodynamic effect of JNJ-42756493 on fibroblast growth factor receptor (FGFR) signaling pathway in tumor. Part 3 is the first Dose Expansion Phase, which is designed to evaluate inclusion biomarkers and preliminary clinical activity at the first RP2D. It consists of 4 expansion cohorts, 1 each for squamous cell lung cancer, small cell lung cancer, breast cancer, other solid tumors (Cohorts A, B, C, and D). Part 4 is the second Dose Expansion Phase, which is designed to evaluate inclusion biomarkers and preliminary clinical activity at the second RP2D. Biomarker eligibility has also been refined based on emerging data. It consists of 2 expansion cohorts, Cohort E for non-small cell lung cancer and Cohort F for select solid tumors including breast, urothelial, GBM, ovarian, head & neck, esophageal, gastric, and cholangiocarcinoma (Cohorts E and F). Enrollment of some cohorts may be discontinued due to lack of enrollment or for futility. The study is estimated to take approximately 48 months to complete. Participants' safety will be monitored throughout the study.