I just came to this conjecture (proved by M.Raynaud and D.Harbater in 1994) last weekend, in Fresnel and v.d.Put's book Rigid Geometry and Its Applications. It claims that all quasi p-group G could be characterized as certain Galois group of a Galois covering Y to P_1, only ramified at infinity (over algebraic closed field with positive character). Since many grandmasters have researched this conjecture, what is the importance of Abhyankar's conjecture ? Well, of course it relates to the inverse Galois theory...