Species' invasions have long been regarded as enormously complex processes, so complex as to defy predictivity. Phases of this process, however, are emerging as highly predictable: the potential geographic course of an invasion can be anticipated with high precision based on the ecological niche characteristics of a species in its native geographic distributional area. This predictivity depends on the premise that ecological niches constitute long-term stable constraints on the potential geographic distributions of species, for which a sizeable body of evidence is accumulating. Hence, although the entire invasion process is indeed complex, the geographic course that invasions are able to take can be anticipated with considerable confidence.