The Riemann Hypothesis is one of the most important unsolved problems in contemporary mathematics. It originates from the German mathematician Georg Friedrich Bernhard Riemann, who proposed it in 1859. The significance of the Riemann Hypothesis lies in the connection between complex analysis and analytic number theory. In the special case for curves over finite fields, we can prove the hypothesis. This proof was provided by André Weil in 1940. In 1973, it was revisited by Enrico Bombieri, who simplified the proof and primarily used techniques from algebraic geometry. The special case of the hypothesis also has important implications. With its help, estimates can be derived regarding the number of rational points on curves over finite fields, which are used, for example, in optimizing geometric codes. Furthermore, this special case was significant for the proof of Fermat's Last Theorem. This book compiles the necessary foundations for proving the Riemann Hypothesis for curves over finite fields and provides a more detailed and elementary proof of the special case.