Species invasions provide numerous unplanned and frequently, but imperfectly, replicated experiments that can be used to better understand the natural world. Classic studies by Darwin, Grinnell, Elton and others on these species-invasion experiments provided invaluable insights for ecology and evolutionary biology. Recent studies of invasions have resulted in additional insights, six of which we discuss here; these insights highlight the utility of using exotic species as 'model organisms'. We also discuss a nascent hypothesis that might provide a more general, predictive understanding of invasions and community assembly. Finally, we emphasize how the study of invasions can help to inform our understanding of applied problems, such as extinction, ecosystem function and the response of species to climate change.