Enhancer elements in the human genome control how genes are expressed in specific cell types and harbor thousands of genetic variants that influence risk for common diseases1-4. Yet, we still do not know how enhancers regulate specific genes, and we lack general rules to predict enhancer-gene connections across cell types5,6. We developed an experimental approach, CRISPRi-FlowFISH, to perturb enhancers in the genome, and we applied it to test >3,500 potential enhancer-gene connections for 30 genes. We found that a simple activity-by-contact model substantially outperformed previous methods at predicting the complex connections in our CRISPR dataset. This activity-by-contact model allows us to construct genome-wide maps of enhancer-gene connections in a given cell type, on the basis of chromatin state measurements. Together, CRISPRi-FlowFISH and the activity-by-contact model provide a systematic approach to map and predict which enhancers regulate which genes, and will help to interpret the functions of the thousands of disease risk variants in the noncoding genome.