In this thesis, we prove a generalization of the classic Gauss-Bonnet theorem, which establishes a surprising link between the curvature and the topology of an orientable closed smooth surface. It turns out that a true generalization of this theorem is only possible in even dimensions, however the methods that work in dimension $2$ are far from being enough to prove this more general statement. To this end, a vast majority of this work is dedicated to preparation for the proof of the main theorem, and during this, a lot of interesting theory that is useful in a broader context is developed. For example, we develop the basics of Riemannian geometry, learn about connections in the more modern environment of principal bundles, and touch upon the theory of characteristic classes when we define the Euler class in the context of smooth vector bundles. Here we also discuss another classic theorem: the Poincaré-Hopf index theorem, which is, in fact, of crucial importance in the proof of the main theorem.
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