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 Universitat Graz, Austria) where we see how far we can stretch methods from random walk theory to get the most explicit results.