The single most important number in the accounts of a non-life insurance company is likely to be the estimate of the outlying liabilities. Since non-life insurance is a major part of our financial industry amounting to up to 5% of BNP in western countries), it is perhaps surprising that mathematical statisticians and experts of operational research (the natural experts of the underlying problem) have left the intellectual work on estimating this number to actuaries. This paper establishes this important problem in a vocabulary accessible to experts of operations research and mathematical statistics and it can be seen as an open invitation to these two important groups of scholars to join this research. The paper introduces a number of new methodologies and approaches to estimating outstanding liabilities in non-life insurance. In particular it reformulates the classical actuarial technique as a histogram type of approach and improves this classical technique by replacing this histogram by a kernel smoother