We discuss some classical problems from insurance mathematics. We
assume that we are looking at an homogeneous portfolio where the number of
claims over the time period till t is denoted by N(t) while the claim sizes are
considered to form a sample from a distribution F. We assume that claim times
and claim sizes are independent. We will give an overview of results dealing
with the total claim amount, with ruin problems in infinite and finite time and
with reinsurance.

We will pay special attention to the differences that occur when the
distribution F has either an exponentially bounded tail or when it is
sub-exponential. This first case treats the situation where the claims are
considered to be light-tailed while the second covers instances where claims
are heavy-tailed. Special reference will be made to recent work with Hansjörg
Albrecher (Technische Universität Graz, Austria) where we see how far we can
stretch methods from random walk theory to get the most explicit results.