Advances in sequencing technologies have enabled the comparison of high-quality genomes of diverse primate species, revealing vast amounts of divergence due to structural variation. Given their large size, structural variants (SVs) can simultaneously alter the function and regulation of multiple genes. Studies estimate that collectively more than 3.5% of the genome is divergent in humans versus other great apes, impacting thousands of genes. Functional genomics and gene-editing tools in various model systems recently emerged as an exciting frontier - investigating the wide-ranging impacts of SVs on molecular, cellular, and systems-level phenotypes. This review examines existing research and identifies future directions to broaden our understanding of the functional roles of SVs on phenotypic innovations and diversity impacting uniquely human features, ranging from cognition to metabolic adaptations.