Structural variants (SVs) are an important source of human genetic diversity but their contribution to traits, disease, and gene regulation remains unclear. The Genotype-Tissue Expression (GTEx) project presents an unprecedented opportunity to address this question due to the availability of deep whole genome sequencing (WGS) and multi-tissue RNA-seq data from 147 individuals. We used comprehensive methods to identify 24,157 high confidence SVs, and mapped cis expression quantitative trait loci (eQTLs) in 13 tissues via joint analysis of SVs, single nucleotide (SNV) and short insertion/deletion (indel) variants. We identified 24,801 eQTLs affecting the expression of 10,101 distinct genes. Based on haplotype structure and heritability partitioning, we estimate that SVs are the causal variant at 3.3-7.0% of eQTLs, which is nearly an order of magnitude higher than prior estimates from low coverage WGS and represents a 26- to 54-fold enrichment relative to their scarcity in the genome. Expression-altering SVs also have significantly larger effect sizes than SNVs and indels. We identified 787 putatively causal SVs predicted to directly alter gene expression, most of which (88.3%) are noncoding variants that show significant enrichment at enhancers and other regulatory elements. By evaluating linkage disequilibrium between SVs, SNVs and indels, we nominate 49 SVs as plausible causal variants at published genome-wide association study (GWAS) loci. Remarkably, 29.9% of the common SV-eQTLs are not well tagged by flanking SNVs, and we observe a notable abundance (relative to SNVs and indels) 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 both common and rare variant association studies.