Structural variants (SVs) are an important source of human genetic diversity, but their contribution to traits, disease and gene regulation remains unclear. We mapped cis expression quantitative trait loci (eQTLs) in 13 tissues via joint analysis of SVs, single-nucleotide variants (SNVs) and short insertion/deletion (indel) variants from deep whole-genome sequencing (WGS). We estimated that SVs are causal at 3.5-6.8% of eQTLs - a substantially higher fraction than prior estimates - and that expression-altering SVs have larger effect sizes than do SNVs and indels. We identified 789 putative causal SVs predicted to directly alter gene expression: most (88.3%) were noncoding variants enriched at enhancers and other regulatory elements, and 52 were linked to genome-wide association study loci. We observed a notable abundance of rare high-impact SVs associated with aberrant expression of nearby genes. These results suggest that comprehensive WGS-based SV analyses will increase the power of common- and rare-variant association studies.