Phylogenetic networks provide a framework for representing complex evolutionary histories involving processes such as hybridization and horizontal gene transfer. An important question in their study is whether the structure of a network (or which features of it) can, in principle, be recovered from genetic data. This talk discusses the identifiability of phylogenetic networks under the multispecies coalescent model. First, I review foundational results for a 'simple' class of networks, then present recent work that establishes identifiability for a much broader network class.