Code-based cryptosystems are promising candidates for post-quantum cryptography, such as the McEliece and Niederreiter cryptosystems. Their security is based on the difficulty of decoding a general linear code. In this thesis, we study the security of algebraic geometry codes in code-based cryptography. These codes include very well-known families like generalized Reed-Solomon, BCH and Goppa codes. Furthermore, they can be described using certain tools, e.g. the theory of algebraic function fields, which we will focus on.