The absolute Galois group G_K of a field K is the Galois group of the separable algebraic closure K^{sep} of K over K. It is the inverse limit of the Galois groups of all finite Galois extensions of K (inside K^{sep}), hence a profinite group. G_K is universal in the sense that the (finite or infinite profinite) groups realizable as Galois groups over K are exactly the quotients of G_K. Very little is known about the question which profinite groups occur as absolute Galois groups. In particular, there is no group theoretic characterization (not even conjecturally) of the profinite groups realizable as G_K's. The scattered results obtained so far suggest that very few profinite groups qualify, that absolute Galois groups tend to be free (or projective) and that deviations from this tendency are typically caused by "local" phenomena like orderings or valuations. We shall recall the classical results in this direction, sketch recent developments and present various precise conjectures.