Extreme climatic events are likely to become more frequent owing to global warming. This may put additional stress on critical infrastructures with typically long life spans. Yet, little is known about the risks of multiple climate extremes on critical infrastructures at regional to continental scales. Here we show the evolution of single- and multi-hazard damages, to energy, transport, industrial and social critical infrastructures in Europe until the year 2100 in view of climate change. We combine a set of high-resolution climate hazard projections, a detailed representation of sectorial physical assets, their sensitivity to the hazards and more than 1100 records of losses from climate extremes in a prognostic modelling framework. We find that damages could triple by the 2020s, multiply six-fold by mid-century, and amount to more than 10 times present damages of €3.4 billion/year by the end of the century. Damages from heat waves, droughts in southern Europe and coastal floods show the most dramatic rise, but the risks of inland flooding, windstorms and forest fires will also increase in Europe, with varying degrees of change across regions. Economic losses are highest for the industry, transport and energy sectors. Future losses will not be incurred equally across Europe. Southern and south-eastern European countries will be most affected and, by association, will likely require higher costs of adaptation. The findings of this study could aid in prioritizing regional investments to address the unequal burden of impacts and differences in adaptation capacities across Europe.