The need to restrict global mean temperature to avoid irreversible climate change is supported by scientific evidence. The need became political practice at the Conference of the Parties in 2015, where the participants decided to limit global warming to not more than +2.0 °C compared to pre-industrial times and to rather aim for a limit of +1.5 °C global warming. Nevertheless, a clear picture of what European climate would look like under +1.5 °C, +2.0 °C and +3.0 °C global warming level (GWL) is still missing. In this study, we will fill this gap by assessing selected climate indices related to temperature and precipitation extremes, based on state of the art regional climate information for Europe taken from the European branch of the World Climate Research Program Coordinated Regional Downscaling Experiment (EURO-CORDEX) ensemble. To assess the impact of these indices under climate change, we investigate the spatial extent of the area of the climate change signal in relation to the affected population. This allows us to demonstrate which climate extremes could be avoided when global warming is kept well below +2.0 °C or even +1.5 °C compared to higher GWLs. The European north–south gradient of tropical nights and hot days is projected to be intensified with an increasing global warming level. For precipitation-related indices, an overall increase in precipitation extremes is simulated, especially under +3.0 °C GWL, for mid- and northern Europe, whereas an increase in dry days is projected for many regions in southern Europe. The benefit of staying below +1.5 °C GWL compared to +2.0 °C GWL is the avoidance of an additional increase in tropical nights and hot days parallel to an increase in dry days in parts of southern Europe as well as an increase in heavy precipitation in parts of Scandinavia. Compared to +3.0 °C GWL, the benefit of staying at +1.5 °C GWL is to avoid a substantial increase (i.e., an increase of more than five dry days and ten tropical nights) in dry days and tropical nights in southern European regions, while, in several European regions and especially in northern Europe, a substantial increase (i.e., more than two heavy precipitation days) in heavy precipitation days could be avoided. This study shows that a statistically significant change in the investigated climate indices can be avoided under +1.5 °C GWL compared to the investigated higher GWLs +2.0 °C and +3.0 °C for the majority of the population in almost all regions. Future studies will investigate compound events where the severity of single extreme events is intensified.