This thesis discusses Levy processes and Levy copulas. In connection with Levy processes we treat some of the theory behind infinitely divisible distributions, acknowledging that the two classes are equivalent.Within the class of Levy processes we will mostly look at stable processes and compound Poisson processes. The theory of Levy processes dates back to the late 1920’s, after de Finetti first introduced the class of infinitely divisible distributions. Since then Levy processes have become popular tools for modelling in finance, insurance and physics. Levy copulas were introduced by Peter Tankov in 2003 in order to model dependency between different components of a multivariate Levy process. In the last part of the book we present an application of Levy copulas in non-life insurance and ruin theory of a Levy copula. Through this example we will discuss aspects regarding estimation of the parameters and goodness of fit.

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