This paper contains little which can be considered as new. It gives a survey of results which have been presented over the last 10-15 years. At one time these results seemed very promising, but in retrospect it is doubtful if they have fulfilled the expectations they raised. In this situation it may be useful to retrace one's steps and see if problems can be reformulated or if new approaches can be found.

Mathematical models have been used in insurance for a long time. One of the first was the Gompertz mortality law; a more recent model, which has been intensively studied is the Compound Poisson Distribution in Lundberg's risk theory.

When a model is introduced, one usually proceeds by stages. The first step is to see if the model appears acceptable on a priori reasons. If it does, the second step is to examine the implications of the model, to see if any of these are in obvious contradiction with observations. If the result of this examination is satisfactory, the third step is usually a statistical analysis to find out how well the model approximates the situation in real life, which one wants to analyse. If the model passes this second examination, the next and final step may be to estimate the parameters of the model, and use it in practice, i.e. to make decisions in the real world.

The advantage of working with a model is that it gives an overall purpose to the collection and analysis of data. A good model should tell us which data we need, and why.