Estimating the risk of single occurences of natural hazards has become impor- tant in recent decades, but up until now it has been largely limited to re-using catalogs of historical events, which usually do not exceed 40 to 50 years in length, and to numerical models, which require heavy computation and are often un- reliable for extrapolation. Extreme value theory provides statistical methods for estimating the frequency of past extreme events as well as for extrapolating beyond observed severities, but natural hazards cannot be modelled using only univariate results. We present an extension of peaks-over-threshold analysis to functions which allows one to de_ne complex extreme events as special types of exceedances, and then obtain their limit distribution for increasingly high thresholds, namely the generalized r-Pareto process. We focus on a speci_c model based on log- Gaussian random functions using classical covariance structures to characterize extremal dependence. Then, we describe a stochastic weather generator for extreme events, capable of quantifying the recurrence of past events as well as generating completely new ones. The methodology is applied to several natural hazards such as windstorms and heavy rainfall. This is joint work with Anthony Davison.