The general theme of this thesis is algebraic function fields. In particular, a focus is put on methods for describing algebraic function fields and their properties using the theory of Gröbner bases. Some necessary basic results on function fields, Gröbner bases and commutative algebra are covered in the first
chapter, and the following chapter derives the results in question. Specifically, it is shown that any function field can be described as an ideal that has a Gröber basis with certain properties. Furthermore, an algorithm for constructing ideals of this kind is given, and way to compute the number of rational places of a function field is proven. The last chapter is devoted to experiments using the theory of the second chapter, and are the first of their kind.