With the onset of the COVID-19 pandemic, there has been an understandable increase in interest in the mathematical and statistical modeling of infectious disease epidemics. The modeling of infectious disease spread through a population generally requires the use of non-standard statistical models. This is primarily because infection events depend upon the infection status of other members of the population (i.e. we cannot assume independence of infection events). Typically, statistical inference for these models (e.g., parameter estimation) is done in a Bayesian context using computational techniques such as Markov chain Monte Carlo (MCMC). This is in part due to the non-standard form of the models, but also in part due to the fact that we often have missing or uncertain data; for example, infection times are rarely observed. Bayesian data augmentation provides a natural framework for allowing for such uncertainty. Further complication is added by the fact that there are often complex heterogeneities in the population which we wish to account for, since, for example, populations do not tend to mix homogeneously. Sometimes, we may wish to account for such heterogeneities using spatial mechanisms that assume that transmission events are more likely to occur between individuals close together in space than individuals further apart. Sometimes, it is more natural to model such heterogeneities using contact networks that represent, for example, the sharing of supplier companies between farms. Here, we will examine the main characteristics of both population-level and individual-level infectious disease models and how they can be fitted to data in a Bayesian MCMC framework.