While many disease-associated variants have been identified through genome-wide association studies, their downstream molecular consequences remain unclear. To identify these effects, we performed cis- and trans-expression quantitative trait locus (eQTL) analysis in blood from 31,684 individuals through the eQTLGen Consortium. We observed that cis-eQTLs can be detected for 88% of the studied genes, but that they have a different genetic architecture compared to disease-associated variants, limiting our ability to use cis-eQTLs to pinpoint causal genes within susceptibility loci. In contrast, trans-eQTLs (detected for 37% of 10,317 studied trait-associated variants) were more informative. Multiple unlinked variants, associated to the same complex trait, often converged on trans-genes that are known to play central roles in disease etiology. We observed the same when ascertaining the effect of polygenic scores calculated for 1,263 genome-wide association study (GWAS) traits. Expression levels of 13% of the studied genes correlated with polygenic scores, and many resulting genes are known to drive these traits.