The organization of the genome in the nucleus and the interactions of genes with their regulatory elements are key features of transcriptional control and their disruption can cause disease. Here we report a genome-wide method, genome architecture mapping (GAM), for measuring chromatin contacts and other features of three-dimensional chromatin topology on the basis of sequencing DNA from a large collection of thin nuclear sections. We apply GAM to mouse embryonic stem cells and identify enrichment for specific interactions between active genes and enhancers across very large genomic distances using a mathematical model termed SLICE (statistical inference of co-segregation). GAM also reveals an abundance of three-way contacts across the genome, especially between regions that are highly transcribed or contain super-enhancers, providing a level of insight into genome architecture that, owing to the technical limitations of current technologies, has previously remained unattainable. Furthermore, GAM highlights a role for gene-expression-specific contacts in organizing the genome in mammalian nuclei.