Spatial pattern, how it arises and how it is maintained, are central foci for ecological theory. In recent years, some attention has shifted from continuum models to spatially discrete analogues, which allow easy treatment of local stochastic effects and of non-local spatial influences. Many of these fall within the area of mathematics known as `interacting particle systems', which provides a body of results that facilitate the interpretation of the suite of simulation models that have been considered, and point towards future analyses. In this paper we review the basic mathematical literature. Three influential examples from the ecological literature are considered and placed within the general framework, which is shown to be a powerful one for the study of spatial ecological interactions.