The folding of genomic DNA from the beads-on-a-string-like structure of nucleosomes into higher-order assemblies is crucially linked to nuclear processes. Here we calculate 3D structures of entire mammalian genomes using data from a new chromosome conformation capture procedure that allows us to first image and then process single cells. The technique enables genome folding to be examined at a scale of less than 100 kb, and chromosome structures to be validated. The structures of individual topological-associated domains and loops vary substantially from cell to cell. By contrast, A and B compartments, lamina-associated domains and active enhancers and promoters are organized in a consistent way on a genome-wide basis in every cell, suggesting that they could drive chromosome and genome folding. By studying genes regulated by pluripotency factor and nucleosome remodelling deacetylase (NuRD), we illustrate how the determination of single-cell genome structure provides a new approach for investigating biological processes.