Gauss-Bonnet 定理是一個美麗的定理，它把曲面上的曲率和曲面的尤拉特徵數做一個連結。換句話說，Gauss-Bonnet 定理是幾何和拓樸之間的橋樑。在本論文中，我們提出 Gauss-Bonnet 定理的發展及証明，並討論它的一些應用。例如，龐加萊-霍普夫指標定理，毛球定理，和代數基本定理。除此之外，我們還討論 $\mathbb{R}^{3}$ 空間中多面體的離散型 Gauss-Bonnet 定理。 Gauss Bonnet theorem is beautiful because it relates the curvature of a surface with its Euler characteristic.It links differential geometry with topology.In this paper, we present some developments on the proof and some applications of Gauss-Bonnet theorem.For example, the Poincar\'{e}-Hopf index theorem, the hairy ball theorem, and the fundamental theorem of algebra.Moreover, we discuss the discrete Gauss-Bonnet theorem about a convex polyhedron in $\mathbb{R}^{3}$.
