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The goal of this book is to rewrite and extend chapter 5 (An Approach to Extremes via Point Processes) of the book "Modelling Extremal Events for Insurance and Finance" by P. Embrechts, C. Klüppelberg and T. Mikosch. The book should result in a text that can be understood from any masters degree student in mathematics, without having large prior knowledge in the area of extreme value theory and point processes. First we introduce some theory that is needed throughout the book. Then we give the definition of a point process and some basic notions about it. In the next chapter we look closer at weak convergence of point processes and present some important theorems, which we use later to derive various limit results. We consider then the point process of exceedances for i.i.d. and strictly stationary sequences. Further we give an alternate approach to results of classical extreme value theory using point process methods, before we move on to consider some linear processes. In the last chapter we present some applications for insurance and finance, which are related to extreme value theory.