European heatwaves at the 0.5, 1.0, 1.5 and 2.0°C global warming levels are examined using bias-corrected daily-mean temperature data from 60 simulations performed with 25 global climate models (GCMs). A heatwave event is defined to consist of at least three consecutive days above the 90th percentile of summer daily-mean temperatures and the event is terminated by two days below the threshold temperature.At the 2.0°C global warming level compared with 0.5°C,the multi-GCM mean annual count of heatwave days is projected to be 3--4-fold in northern and more than six-fold in southern Europe. The total annual heatwave extremity index, or the sum of exceedings above the threshold temperature over all the heatwave days, becomes approximatively four-fold in the north and ten-fold in the south. In central Europe, the date of the strongest heatwave of year is delayed by about one week.By concatenating the bias-corrected output data of all the GCM runs, 1200-year samples can be created for all four warming levels, from which the return levels of the strongest heatwaves can be determined very robustly. An intense heatwave occurring once in 10 years at the 0.5°C warming level has an annual probability of about 50% in northern and 80-90% in southern Europe under 2.0°C. Corresponding probabilities for 100-year heatwaves would be nearly 20% in the north and about 60% in the south.Finally, we discuss factors that explain the stronger increase in heatwave duration and extremity in the south than north.