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Genome-wide association studies (GWAS) have identified thousands of noncoding loci that are associated with human diseases and complex traits, each of which could reveal insights into the mechanisms of disease. Many of the underlying causal variants may affect enhancers, but we lack accurate maps of enhancers and their target genes to interpret such variants. We recently developed the activity-by-contact (ABC) model to predict which enhancers regulate which genes and validated the model using CRISPR perturbations in several cell types. Here we apply this ABC model to create enhancer-gene maps in 131 human cell types and tissues, and use these maps to interpret the functions of GWAS variants. Across 72 diseases and complex traits, ABC links 5,036 GWAS signals to 2,249 unique genes, including a class of 577 genes that appear to influence multiple phenotypes through variants in enhancers that act in different cell types. In inflammatory bowel disease (IBD), causal variants are enriched in predicted enhancers by more than 20-fold in particular cell types such as dendritic cells, and ABC achieves higher precision than other regulatory methods at connecting noncoding variants to target genes. These variant-to-function maps reveal an enhancer that contains an IBD risk variant and that regulates the expression of PPIF to alter the membrane potential of mitochondria in macrophages. Our study reveals principles of genome regulation, identifies genes that affect IBD and provides a resource and generalizable strategy to connect risk variants of common diseases to their molecular and cellular functions.