This article considers stochastic models for the spread of an infection in a structured community, where this structured community is itself described by a random network model. Some common network models and transmission models are defined and large population properties of them are presented. The focus is then shifted to statistical methodology: what can be estimated and how, depending on the underlying network, transmission model, and the available data? This survey article discusses several different scenarios, giving references to publications where more details can be found, and identifies important open problems.