Brief Summary: This phase II trial studies how well brigatinib works in treating patients with ALK and ROS1 gene alterations and solid cancers that have spread to nearby tissue and lymph nodes or other places in the body. Brigatinib may stop the growth of tumor cells by blocking some of the enzymes needed for cell growth.

Detailed Description: PRIMARY OBJECTIVES: I. To evaluate the overall response rate (ORR) of brigatinib in patients with advanced solid tumors harboring genomic alterations in ALK (excluding lung) and ROS1 (all solid tumors). SECONDARY OBJECTIVES: I. To assess the safety and tolerability of brigatinib in patients with advanced solid tumors harboring genomic alterations in ALK (excluding lung) and ROS1 (all solid tumors). II. To assess progression free survival (PFS) and overall survival (OS) in patients with advanced ALK or ROS1 mutated solid tumors treated with brigatinib. III. To assess sensitivity and durability of response to brigatinib in different solid tumor types. IV. To assess the role of intertumoral and intratumoral heterogeneity in the development of resistance to brigatinib. V. To identify candidate genomic (including circulating tumor deoxyribonucleic acid \[DNA\]) and proteomic biomarkers of tumor sensitivity and resistance to brigatinib using high-throughput approaches (exome, transcriptome, reverse phase protein array \[RPPA\]). TERTIARY OBJECTIVES: I. Correlation of brigatinib exposure with efficacy and safety. II. Correlation of tumor and plasma biomarkers with brigatinib efficacy and safety. OUTLINE: Patients receive brigatinib orally (PO) once daily (QD) on days 1-28. Courses repeat every 28 days in the absence of disease progression or unacceptable toxicity. After completion of study treatment, patients are followed up every 3 months for 52 weeks.