In populations with cyclic dynamics, invasions typically generate either regular (periodic) spatiotemporal oscillations, spatiotemporal chaos, or a mixture of the two. I will present examples of these behaviours in numerical simulations of invasions. I will then describe how to determine which of the behaviours will occur, as a function of ecological parameters. The characteristic invasion profile is a band of regular oscillations immediately behind the invasion front, which undergoes a subsequent transition to chaos. The key determinant of long-term behaviour is the width of the band, and I will explain how to calculate it. Specific applications of the results reveal a marked sensitivity to ecological parameters. In the light of recent evidence that climate change is having a significant effect on the demographic parameters underlying oscillatory ecological systems, this implies a potentially dramatic effect of climate change on the spatiotemporal dynamics generated by invasions.