Several years ago, the speaker and M.Raynaud proved Abhyankar's 1957 conjecture concerning Galois groups over affine curves in characteristic p. Since then, Abhyankar has proposed conjectures to handle the higher dimensional case, as well as the case of curves over finite fields. The purpose of this talk is to show that the higher dimensional conjecture does not in general hold as stated. The case of curves over finite fields will also be discussed.