Here, we review the growing body of research concerning the spread of infectious diseases on networks, focusing on the interplay between network theory and epidemiology. The paper is split into four main sections which examine the types of network relevant to epidemiology, the multitude of ways these networks can be characterised, the statistical methods that can be applied to either infer the likely network structure or the epidemiological parameters on a realised network, and finally simulation and analytical methods to determine epidemic dynamics on a given network. Given the breadth of areas covered and the ever-expanding number of publications (over seven thousand papers have been published concerning infectious diseases and networks) a comprehensive review of all work is impossible. Instead, we provide a personalised overview into the areas of network epidemiology that have seen the greatest progress in recent years or have the greatest potential to provide novel insights. As such considerable importance is placed on analytical approaches and statistical methods which are both rapidly expanding fields. We note that a range of other networkbased processes (such as the spread of ideas or panic) can be modelled in a similar manner to the spread of infection; however, in these contexts, the transmission process is far less clear; therefore, throughout this paper, we restrict our attention to epidemiological issues.