Globalisation has led to the introduction of thousands of alien species. Understanding the invasion process is critical to predicting impacts and informing management. Invasion dynamics have been assumed to follow an ‘invasion curve’ (S-shaped curve of available area invaded), and similarly, we propose an ‘impact curve’ describing the impacts generated by invasive alien species over time based on cumulative abundance. To test this curve’s large-scale applicability, we used the New Zealand mud snail Potamopyrgus antipodarum, one of the most damaging freshwater invaders. Using long-term (1979-2020) abundance and environmental data across 306 European sites, we observed that P. antipodarum abundance generally increased through time, with slower population growth at higher latitudes and with lower runoff depth. Fifty-nine percent of these populations followed the impact curve, characterized by first occurrence, exponential growth, then long-term saturation. This behaviour is consistent with boom-bust dynamics, as saturation occurs due to a rapid decline in abundance over time. Across sites, we estimated that impact peaked approximately two decades after first detection. The S-shaped impact curve may be common among many invasive species that undergo complex invasion dynamics, urging rapid management actions to mitigate impact.