Spaces of phylogenetic trees have been studied for several years. As well as having mathematical and computational interest, they also have central importance within various tree-building algorithms. Recently, there has been growing interest in defining spaces of phylogenetic networks. Much less is known about such spaces. Indeed, it is often significantly more complex to understand them, as well to develop network building algorithms that exploit their structure. In this talk, we shall review some of the network spaces that have been recently defined, introduce some new ones, and discuss some of the challenges associated with putting them to practical use.