Biologists have recently devoted increasing attention to the role of rapid evolution in species' responses to environmental change. However, it is still unclear what evolutionary responses should be expected, at what rates, and whether evolution will save populations at risk of extinction. The potential of biological invasions to provide useful insights has barely been realised, despite the close analogies to species responding to global change, particularly climate change; in both cases, populations encounter novel climatic and biotic selection pressures, with expected evolutionary responses occurring over similar timescales. However, the analogy is not perfect, and invasive species are perhaps best used as an upper bound on expected change. In this article, we review what invasive species can and cannot teach us about likely evolutionary responses to global change and the constraints on those responses. We also discuss the limitations of invasive species as a model and outline directions for future research.