The concept of Gröbner bases, introduced by Bruno Buchberger in 1965, is of distinguished importance in symbolic computation. It provides an algorithmic solution to a variety of questions related to systems of polynomial equations (in several variables) which cannot be solved by other means.
In the course, we give a thorough introduction into the theoretical background of Gröbner bases. We discuss improvements and generalizations of Buchberger's algorithm for computing Gröbner bases. We present algorithms relying themselves on Gröbner basis computations. If time permits, we will also see some ``real world applications'' of Gröbner bases and talk about ongoing research topics.