Coupled general circulation models are of paramount importance to quantitatively assessing the magnitude of future climate change. Usual methods for validating climate models include the evaluation of mean values and covariances, but less attention is directed to the evaluation of extremal behaviour. This is a problem because many severe consequences of climate change are due to climate extremes. We present a method for model validation in terms of extreme values based on classical extreme value theory. We further discuss a clustering algorithm to detect spatial dependencies and tendencies for concurrent extremes. To illustrate these methods, we analyse precipitation extremes of the Alfred Wegener Institute Earth System Model (AWI-ESM) global climate model and from other models that take part in the Coupled Model Intercomparison Project CMIP6 and compare them to the reanalysis data set CRU TS4.04. The clustering algorithm presented here can be used to determine regions of the climate system that are then subjected to a further in-depth analysis, and there may also be applications in palaeoclimatology.