This note surveys recent developments in the problem of computing Galois groups. Galois theory stands at the cradle of modern algebra and interacts with many areas of mathematics. The problem of determining Galois groups therefore is of interest not only from the point of view of number theory (for example see the article [39] in this volume), but leads to many questions in other areas of mathematics. An example is its application in computer algebra when simplifying radical expressions [32] .

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... ot surprisingly, this task has been considered in works from number theory, group theory and algebraic geometry. In this note I shall give an overview of methods currently used. While the techniques used for the identification of Galois groups were known already in the last century [26], the involved calculations made it almost impractical to do computations beyond trivial examples. Thus the problem was only taken up again in the last 25 years with the advent of computers. In this note we will restrict ourselves to the case of the base field Q. Most methods generalize to other fields like Q(t), Q p , IF p (t) or number fields. The results presented here are the work of many mathematicians. I tried to give credit by references wherever possible.