| This talk deals with the asymptotic behavior of Lévy processes.
More precisely, the probability tails of suprema over compact intervals are
studied. Lévy processes are divided into a handfull of classes, depending
on the weight of the tails of their univariate marginal distributions. Different
methods are used for the different classes. The methods we use include generalizations
of known techniques, as well as completely new techniques, developed by us.
The result is a quite complete treatment of the mentioned asymptotic problem.
Several of the processes, the asymptotics of which are here studied for
the first time, have recently become important in the field of mathematical
finance. This means that our results could have impact on, for example,
the assesments of financial risk.