Insurance is the business dedicated to indemnifying random financial losses in exchange of charging premiums. One can also regard it as an attempt to tame the uncertainty which underlies many aspects of human existence. These risks can be of very different nature, but the classical division is made between life insurance and non-life insurance. The work presented in this talk is dedicated to various aspects of the latter, mainly through the specification of new probabilistic models. These models not only give insight into the nature of the possible drivers behind the uncertain events in insurance, but often lead to a statistical approach which can be applied in practice. The main themes throughout are the analysis of ood data, the mathematical modeling of the surplus and dividends of an insurance company, and the statistical modeling of claim severities. While the challenges addressed in this thesis are motivated by those insurance applications, the developed methods may also be interesting in their own right, and for other application areas. In the following sections of this chapter, we will expand further on the topics covered in the remainder of the thesis.