It is now increasingly common for infectious disease epidemics to be analysed with mathematical models. Modelling is possible because epidemics involve relatively simple processes occurring within large populations of individuals. Modelling aims to explain and predict trends in disease incidence, prevalence, morbidity or mortality. Models give important insight into the development of epidemics. Following disease establishment, epidemic growth is approximately exponential. The rate of growth in this phase is primarily determined by the basic reproduction number (R0), the number of secondary cases per primary case when the population is susceptible. R0 also determines the ease with which control policies can control epidemics. Once a significant proportion of the population has been infected, not all contacts of an infected individual will be with susceptible people. Infection can now continue only because new births replenish the susceptible population. Eventually, an endemic equilibrium is reached whereby every infected person infects one other individual on average. Heterogeneity in host susceptibility, infectiousness, human contact patterns and the genetic composition of pathogen populations introduces substantial additional complexity into the models required to model real diseases realistically. The contribution concludes with a brief review of the recent application of mathematical models to emerging human and animal epidemics, notably the spread of HIV in Africa, the variant Creutzfeldt-Jakob disease epidemic in the UK and its relationship to bovine spongiform encephalitis in cattle, the 2001 foot and mouth epidemic in UK livestock, bioterrorism threats such as smallpox, and the SARS epidemics in 2003.