Genomic structural variants have long been implicated in phenotypic diversity and human disease, but dissecting the mechanisms by which they exert their functional impact has proven elusive. Recently however, developments in high-throughput DNA sequencing and chromosomal engineering technology have facilitated the analysis of structural variants in human populations and model systems in unprecedented detail. In this Review, we describe how structural variants can affect molecular and cellular processes, leading to complex organismal phenotypes, including human disease. We further present advances in delineating disease-causing elements that are affected by structural variants, and we discuss future directions for research on the functional consequences of structural variants.