Genomes across a wide range of eukaryotic organisms fold into higher-order chromatin domains. Topologically associating domains (TADs) were originally discovered empirically in low-resolution Hi-C heat maps representing ensemble average interaction frequencies from millions of cells. Here, we discuss recent advances in high-resolution Hi-C, single-cell imaging experiments, and functional genetic studies, which provide an increasingly complex view of the genome's hierarchical structure-function relationship. On the basis of these new findings, we update the definitions of distinct classes of chromatin domains according to emerging knowledge of their structural, mechanistic and functional properties.