Abstract: The estimation of quantities related to extreme events is the main issue of this thesis. We find its applications in practice, for example in insurance mathematics, where it is of great importance to have statistical insight into the occurrence of large claims, due to natural catastrophes such as floods, hurricanes, etc. In the univariate setting we are interested in quantities like high quantiles (i.e. for example a level of the sea that is exceeded with a given very low probability), endpoint of a distribution function (e.g. life span of a certain group of individuals) or tail probability (e.g. the probability that a given level of the sea in a certain location is exceeded). We look for the optimal sub-sample size to use in the estimation and related questions. In the bivariate setting, i.e. when we have two variables jointly playing a role, the estimation of tail dependence and probability of an extreme set is addressed.