In this thesis, we study ruin probability functions. In the first part, we define the notion of a Levy process. Also, we discuss certain properties and provide examples of these processes. In particular, we discuss the notion of characteristic function of a distribution function, infinitely divisible and stable distribution functions, characteristic exponent and Lévy measure of an infinitely divisible distribution function. At the end we provide several classical examples of Lévy processes, and compute their Lévy characteristics. In the second part of the thesis, we deal with intersections of ruin probability functions. We discuss the notion of Laplace exponent, Laplace transform and scale function. In particular, we explicitly compute ruin probability of the classical risk process and spectrally negative \(\alpha-\)stable process and discuss intersections between two ruin probability functions. Then we discuss a classical risk model perturbed by a \(\alpha-\)stable spectrally negative Lévy processes. In the end, we state a result which gives sufficient conditions for arbitrary number of intersections of two ruin probability functions.