Abstract: The diffeomorphism classification of 4-manifolds is at the moment hopeless. Only in special cases one can say something, mainly using Kirby calculus. But there is a surprising simplification if one allows stabilization by connected sum with copies of S^2 x S^2. This stable classification can in principle be attacked for arbitrary closed smooth 4-manifolds. The situation is much better for topological 4-manifolds thanks to Freedman’s proof of the topological s-cobordism theorem for „good“ fundamental groups. The one can use surgery techniques to obtain rather concrete theorems for a large class of fundamental groups. In the talk I will concentrate on results which allow a non-specialist to carry out classification result by himself.