Scale transition theory is a mathematical technique for understanding changes in population dynamics with changes in spatial or temporal scale. It explains the emergence of new properties on large scales from the interaction between nonlinearities and variation on small scales. It applies statistical theory for averaging nonlinear functions to understanding this interaction. The fundamental concepts are most easily illustrated with reference to spatial models where state variables on larger spatial scales are simply defined as averages of those on smaller scales. Scale transition theory also explains the conceptually difficult topic of how species coexistence arises from temporal fluctuations. In this case, averages of per capita growth rates over time define long-term population trends and outcomes, and these averages are critically affected by interactions between nonlinear dynamics and temporal variation in state variables and environmental variables. Two general mechanisms of species coexistence, the storage effect and relatively nonlinear competitive variance, emerge.