Aim European colonisation of Australia triggered a cascade of processes that continue to threaten biodiversity today. These include the introduction of invasive species, responsible for rapid and widescale extinctions. Here, we integrated process-based models with thousands of first sighting records to reconstruct the arrival and spread of European red foxes (Vulpes vulpes) in Australia at a high spatiotemporal resolution. Location: Continental Australia. Methods We constructed tens of thousands of spatially explicit population models of fox colonisation dynamics, using historical records, demographic estimates and spatiotemporal patterns of climate and environmental change. We then tested model outputs against inferences of timings of fox arrival from newspaper articles and empirical estimates of population sizes. Results We show that it took just 60 years for foxes to infill their entire potential distribution of Australia. Foxes colonised mesic regions of Australia earlier than the more arid regions of central and north-western Australia, reaching carrying capacity shortly after the middle of the 20th century. Conclusions These new insights into the temporal pattern of spread of foxes across Australia, its underpinning processes, and associated maps of abundances promise to help better explain past losses of biodiversity, providing critical information needed to avert future extinctions of Australia's unique fauna. Our process-based approach for projecting invasion histories should prove similarly useful for reconstructing the biogeography of other invasive species, including cryptic populations at the early stages of invasion.