Three fundamental problems still exist in the study of networks. Firstly, are there relatively low-dimensional ways of capturing key aspects of a network's structure? What constitutes a key aspect will vary with the problem being studied, but for epidemiological applications, it should be hoped that a universal set of network characteristics may emerge. There is then the task of assessing reasonable and realistic ranges for these key variables based on values computed for known transmission networks-unfortunately very few transmission networks have been recorded in any degree of detail although modern electronic devices may simplify the process in the future. Secondly, there is the related statistical problem of inferring plausible complete networks from the partial information collected by methods such as contact tracing. This is equivalent to seeking an underlying model for the network connections that is consistent with the known partial information, and hence, has strong resonance with the more mechanistically motivated models in Section 2.3. Even when the network is fully realised (and an epidemic observed), there is considerable statistical difficulty in attributing risk to particular contact types. Finally, there are the key questions of predicting the dynamics of infection on any given network-and while for many complex networks, direct simulation is the only approach, for other simplified networks some analytical traction can be achieved, which helps to provide more generic insights into which elements of network structure are most important. These three key areas are discussed below.