A key goal of whole-genome sequencing for studies of human genetics is to interrogate all forms of variation, including single-nucleotide variants, small insertion or deletion (indel) variants and structural variants. However, tools and resources for the study of structural variants have lagged behind those for smaller variants. Here we used a scalable pipeline1 to map and characterize structural variants in 17,795 deeply sequenced human genomes. We publicly release site-frequency data to create the largest, to our knowledge, whole-genome-sequencing-based structural variant resource so far. On average, individuals carry 2.9 rare structural variants that alter coding regions; these variants affect the dosage or structure of 4.2 genes and account for 4.0–11.2% of rare high-impact coding alleles. Using a computational model, we estimate that structural variants account for 17.2% of rare alleles genome-wide, with predicted deleterious effects that are equivalent to loss-of-function coding alleles; approximately 90% of such structural variants are noncoding deletions (mean 19.1 per genome). We report 158,991 ultra-rare structural variants and show that 2% of individuals carry ultra-rare megabase-scale structural variants, nearly half of which are balanced or complex rearrangements. Finally, we infer the dosage sensitivity of genes and noncoding elements, and reveal trends that relate to element class and conservation. This work will help to guide the analysis and interpretation of structural variants in the era of whole-genome sequencing.