A statistical extreme value analysis is applied to very high-resolution climate model results and observations encompassing the area of Germany. Two control runs representing the current climate, as well as three scenario simulations of the regional climate model REMO, are investigated. The control runs were compared against high-resolution observations. The analysis is divided into two main parts: first trends in extreme quantiles of daily precipitation totals are estimated in a station-by-station analysis. In the second part, the spatial characteristics of the estimated trends in heavy rainfall are investigated over the area of Germany by fitting a parametric geostatistical model to these trends. The rule of thumb of estimating trends in extreme quantiles of heavy precipitation based on the Clausius-Clapeyron relation, about 6.5% per 1°C temperature increase, has been roughly confirmed for Germany by our study with respect to the observations, but the climate model computes weaker trends. In the control simulations, the climate model tends to underestimate trends in heavy rainfall compared to observations. In the scenario simulations, positive trends prevail (as in the observations). They are, however, relatively small when set in relation to the uncertainties. The trends become significantly positive to a larger spatial extent only in the A2 scenario simulation. The estimated shape of the extreme value distributions does not change significantly in the scenario simulations compared to the climate model control runs. The parameter estimates for the geostatistical model for the trends in extreme quantiles of daily precipitation sums are rather uncertain. The most striking feature of the analysis is a reduction of the spatial variance of the trends over the considered area of Germany in the scenario simulations compared to observations and, in particular, the climate model control runs.