Mammalian genomes harbor millions of noncoding elements called enhancers that quantitatively regulate gene expression, but it remains unclear which enhancers regulate which genes. Here we describe an experimental approach, based on CRISPR interference, RNA FISH, and flow cytometry (CRISPRi-FlowFISH), to perturb enhancers in the genome, and apply it to test >3,000 potential regulatory enhancer-gene connections across multiple genomic loci. A simple equation based on a mechanistic model for enhancer function performed remarkably well at predicting the complex patterns of regulatory connections we observe in our CRISPR dataset. This Activity-by-Contact (ABC) model involves multiplying measures of enhancer activity and enhancer-promoter 3D contacts, and can predict enhancer-gene connections in a given cell type based on chromatin state maps. Together, CRISPRi-FlowFISH and the ABC 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.We defined Activity (A) as the geometric mean of the read counts of DHS and H3K27ac ChIP-Seq at an element E, and Contact (C) as the normalized Hi-C contact frequency between E and the promoter of gene G (see Methods). (The ABC score performed similarly across a range of data preprocessing parameters, and when defining Activity using other combinations of measurements of chromatin accessibility, histone modifications, and nascent transcription, see Methods, Fig. S6,S7,S8).The ABC model performed remarkably well, and much better than alternatives, at predicting DE-G connections in our CRISPR dataset. The quantitative ABC score correlated with the experimentally measured relative effects of candidate elements on gene expression (Spearman ρ for regulatory DE-G pairs = -0.68 Fig. 3C). Binary classifiers based on thresholds on the ABC score substantially outperformed existing predictors of enhancer-gene regulation. For example, when we used an ABC threshold corresponding to 70% recall, the predictions had 63% precision, and the area under precision-recall curve (AUPRC) was 0.66, compared to 0.36 for predictions based solely on genomic distance (Fig. 3A).