Extreme climate events have been investigated by many researchers in recent decades, and statisticians too have developed statistical tools capable of dealing with them. Although extreme value theory has been extensively developed and used in modelling events such as extreme rainfall and heat waves, the spatial nature of climate data requires new tools to deal with spatial extremes. This thesis first reviews existing models and methods for modelling extreme events, and then combines spatial max-stable random processes and composite likelihood to allow likelihood-based inference on spatial extreme data. The properties of these models and estimators are assessed by simulation and a data set on Swiss temperature at 17 sites for 45 years is analysed, with simulation used to make predictions of future extreme events. The same methodology can be employed for extremes in one-dimensional space, and it is illustrated by the analysis of the extremes of a rainfall time series.