Moment closure approximation (MCA) is a method of obtaining dynamic deterministic approximations to models where spatiality is important. Such approximations track the time evolution of low-order correlations, for instance the correlation of disease status of nearest-neighbours in a square lattice. Thus they are able to capture aspects of population dynamics which traditional mean-field approximations are unable to. This thesis extends the techniques of moment closure approximation and develops novel applications for MCA in epidemiology. Most existing moment closures were intended as deterministic approximations to static regular lattices. However we develop deterministic approximations for dynamic network models and continuous space models. The purpose of applying MCA to a different set of models is not only to demonstrate their flexibility; we also explore the dynamical properties of such models with the moment closure tools we derive and with simulation data. Comparisons are then made between processes on regular lattices and processes in dynamic networks and in continuous space. Additionally, we answer questions relating to the epidemiology of sexually transmitted diseases and epidemics in populations embedded in two-dimensional continuous space. Some of the new techniques we develop can be applied to other models in ecology and epidemiology. We conclude that moment closure approximations continue to provide fertile ground for research, and that application of MCA to models other than static regular lattices can be worthwhile. Chapter 1 consists of background material and an introduction to moment closure approximations. In chapter 2 we look at the properties of moment closure approximations near critical points and during transient phases and consider their accuracy in such cases. Chapters 3 and 4 cover the application of pair approximations to sexually transmitted disease models, and chapter 5 is a preliminary study of a pair approximation for a continuous space model.