Structural variants (SVs) underlie important crop improvement and domestication traits. However, resolving the extent, diversity, and quantitative impact of SVs has been challenging. We used long-read nanopore sequencing to capture 238,490 SVs in 100 diverse tomato lines. This panSV genome, along with 14 new reference assemblies, revealed large-scale intermixing of diverse genotypes, as well as thousands of SVs intersecting genes and cis-regulatory regions. Hundreds of SV-gene pairs exhibit subtle and significant expression changes, which could broadly influence quantitative trait variation. By combining quantitative genetics with genome editing, we show how multiple SVs that changed gene dosage and expression levels modified fruit flavor, size, and production. In the last example, higher order epistasis among four SVs affecting three related transcription factors allowed introduction of an important harvesting trait in modern tomato. Our findings highlight the underexplored role of SVs in genotype-to-phenotype relationships and their widespread importance and utility in crop improvement. Display omitted •Long-read sequencing of 100 tomato genomes uncovered 238,490 structural variants•Transposons underlie many SVs, and SV hotspots revealed large introgressions•SVs associated with genes are predictive of population-scale changes in expression•New genome assemblies resolved complex breeding QTLs caused by SVs Comprehensive structural variant identification in tomato genomes allows insight into the evolution and domestication of tomato and serves as a resource for phenotype-directed breeding.