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The biomedical community is increasingly invested in capturing all genetic variants across human genomes, interpreting their functional consequences and translating these findings to the clinic. A crucial component of this endeavour is the discovery and characterization of structural variants (SVs), which are ubiquitous in the human population, heterogeneous in their mutational processes, key substrates for evolution and adaptation, and profound drivers of human disease. The recent emergence of new technologies and the remarkable scale of sequence-based population studies have begun to crystalize our understanding of SVs as a mutational class and their widespread influence across phenotypes. In this Review, we summarize recent discoveries and new insights into SVs in the human genome in terms of their mutational patterns, population genetics, functional consequences, and impact on human traits and disease. We conclude by outlining three frontiers to be explored by the field over the next decade.