本論文主要證明二個在幾何上重要的定理：Gauss-Bonnet定理與Riemann-Roch定理，且與指標定理做一個連結。第一章主要是用陳省身在1943年發表的”內蘊”手法來證明二維流型上的Gauss-Bonnet定理。第二章主要是介紹古典的Riemann- Roch定理，以三種不同的形式給出，並在第三章證明第三種上同調化的形式。第四章是藉由計算二個橢圓算子的指標得到流形上的拓樸不變量，此為Atiyah- Singer指標定理。 In this thesis, we prove two important theorems in geometry. In chapter one, we state the Gauss-Bonnet theorem on even dimensional manifold and give the detail of the proof of two dimensional case. The proof is based on the paper "A simple intrinsic proof of the Gauss-Bonnet formula for closed Riemannian manifold", published by S.S. Chern in 1943. A little history of this theorem is included. Chapter two and three mainly focus on Riemann-Roch theorem on one-dimensional complex manifold, Riemann surface. We establish some basics on Riemann surface in chapter two, such as divisors, holomorphic line bundles, sheaves and cohomology on sheaves, also Hodge theorem in the end of this chapter. The proof of Riemann-Roch is in the chapter three. In chapter four, we show a theorem by calculating two analytic indices of two operators, which give us Gauss-Bonnet and Riemann-Roch theorem. This theorem is the Atiyah-Singer index theorem, proved by Atiyah and Singer in 1963.