The coypu (Myocastor coypus) is a semi-aquatic rodent native to South America which has become invasive in Europe and other parts of the world. Although recently listed as species of European Union concern in the EU Invasive Alien Species Regulation, an analysis of the current European occurrence and of its potential current and future distribution was missing yet. We collected 24,232 coypu records (corresponding to 25,534 grid cells at 5 × 5 km) between 1980 and 2018 from a range of sources and 28 European countries and analysed them spatiotemporally, categorising them into persistence levels. Using logistic regression, we constructed consensus predictions across all persistence levels to depict the potential current distribution of the coypu in Europe and its change under four different climate scenarios for 2041–2060. From all presence grid cells, 45.5% showed at least early signs of establishment (records temporally covering a minimum of one generation length, i.e. 5 years), whereas 9.8% were considered as containing established populations (i.e. three generation lengths of continuous coverage). The mean temperature of the warmest quarter (bio10), mean diurnal temperature range (bio2) and the minimum temperature of the coldest month (bio6) were the most important of the analysed predictors. In total, 42.9% of the study area are classified as suitable under current climatic conditions, of which 72.6% are to current knowledge yet unoccupied; therefore, we show that the coypu has, by far, not yet reached all potentially suitable regions in Europe. Those cover most of temperate Europe (Atlantic, Continental and Pannonian biogeographic region), as well as the coastal regions of the Mediterranean and the Black Sea. A comparison of the suitable and occupied areas showed that none of the affected countries has reached saturation by now. Under climate change scenarios, suitable areas will slightly shift towards Northern regions, while a general decrease in suitability is predicted for Southern and Central Europe (overall decrease of suitable areas 2–8% depending on the scenario). Nevertheless, most regions that are currently suitable for coypus are likely to be so in the future. We highlight the need to further investigate upper temperature limits in order to properly interpret future climatic suitability for the coypu in Southern Europe. Based on our results, we identify regions that are most at risk for future invasions and provide management recommendations. We hope that this study will help to improve the allocation of efforts for future coypu research and contribute to harmonised management, which is essential to reduce negative impacts of the coypu and to prevent further spread in Europe.