The present study describes the climatology of extreme events of surface temperature. Using both data analysis and model simulations, the frequency, duration and structure of extreme events are investigated. After developing objective methods to identify extreme events, these are consistently applied to all data used in the study, focusing mainly on two particular regions: Europe and North America. 54 years (1950-2003) of Reanalysis data are used and the space-time structure of extreme events during 1950-19 and 1977-2003 is compared and contrasted. The study evidences that there has been an increase in warm extremes in Europe and continental Canada and a general decrease of cold spells. These techniques and methods are also applied to model simulations with increasing concentration of greenhouse gases to evaluate how the occurrence of extreme warm and cold events might change in the future: the projections agree in an increase of warm events in the XXI century. The Mediterranean SST anomaly is identified as a common feature for major warm events in the central-western European region. The European heat wave of summer 2003 is then taken as a case study analyzed with GCM simulations.