Genomic DNA is folded into loops and topologically associating domains (TADs), which serve important structural and regulatory roles. It has been proposed that these genomic structures are formed by a loop extrusion process, which is mediated by structural maintenance of chromosomes (SMC) protein complexes. Recent single-molecule studies have shown that the SMC complexes condensin and cohesin are indeed able to extrude DNA into loops. In this Review, we discuss how the loop extrusion hypothesis can explain key features of genome architecture; cellular functions of loop extrusion, such as separation of replicated DNA molecules, facilitation of enhancer-promoter interactions and immunoglobulin gene recombination; and what is known about the mechanism of loop extrusion and its regulation, for example, by chromatin boundaries that depend on the DNA binding protein CTCF. We also discuss how the loop extrusion hypothesis has led to a paradigm shift in our understanding of both genome architecture and the functions of SMC complexes.