Structural variants (SVs) are an important source of human genome diversity but their functional effects are not well understood. We mapped 61,668 SVs in 613 individuals with deep genome sequencing data from the GTEx project and measured their effects on gene expression. We estimate that common SVs are causal at 2.66% of eQTLs, which is a 10.5-fold enrichment relative to their abundance in the genome and consistent with prior work using smaller sample sizes. Duplications and deletions were the most impactful variant types, whereas the contribution of mobile element insertions was surprisingly small (0.12% of eQTLs, 1.9-fold enriched). Multi-tissue analysis of expression effects revealed that gene-altering SVs show significantly more constitutive effects than other variant types, with 62.09% of coding SV-eQTLs active in all tissues with known eQTL activity compared to 23.08% of coding SNV- and indel-eQTLs, whereas noncoding SVs, SNVs and indels show broadly similar patterns. We also identified 539 rare SVs associated with nearby gene expression outliers. Of these, 62.34% are noncoding SVs that show strong effects on gene expression yet modest enrichment at known regulatory elements, demonstrating that rare noncoding SVs are a major source of gene expression differences but remain difficult to predict from current annotations. Remarkably, both common and rare noncoding SVs often show strong regional effects on the expression of multiple genes: SV-eQTLs affect an average of 1.82 nearby genes compared to 1.09 genes affected by SNV- and indel-eQTLs, and 21.34% of rare expression-altering SVs show strong effects on 2-9 different genes. We also observe significant effects on gene expression extending 1 Mb from the SV. This provides a mechanism by which individual noncoding SVs may have strong and/or pleiotropic effects on phenotypic variation and disease.