Invasions by exotic species are occuring more frequently than ever before. The ecological and economic consequenses of these invasions can be dramatic. To cope with invading species we must be able to answer certain key questions: Will a nascent invasion grow and spread? How quickly? What will be effects on other species? How will the invasion change the ecosystem's function? What mitigation strategies are feasible? Which will be most effective? The answers to these questions depend not only on intrinsic characteristics of the invader (its life history and how it disperses), but also on the relationships between these characteristics and the physical environment (e.g. temperature, rainfall, soil characteristics, etc.) and the relationship between the invader and the community it is invading (e.g. its predators, prey, and comptetitors). These relationships are complex---they occur over different temporal and spatial scales, can be stochastic, and are almost invariably nonlinear. As a result, they have usually been ignored. I will present examples of these complexities in natural systems, and report on some recent theoretical advances that begin to incorporate these complexities in a systematic way.