The computation of Galois groups is the only problem out of Zassenhaus's four main problems in computational algebra that still has no general solution.

While substantial progress has been made in the recent years, computation of Galois groups of polynomials of degree up to 23 are now routinely done (using Magma), no degree-independent algorithm has been published so far.

In my talk I will explain what computational challenges need solving to remove the dependence on pre-computed data and report on recent advances in doing so.