The integral cohomology rings of the configuration spaces of $n$-tuples of distinct points on arbitrary surfaces (not necessarily orientable, not necessarily compact and possibly with boundary) are studied. It is shown that for punctured surfaces the cohomology rings stabilize as the number of points tends to infinity, similarly to the case of configuration spaces on the plane studied by Arnold, and the Goryunov splitting formula relating the cohomology groups of configuration spaces on the plane and punctured plane to arbitrary punctured surfaces is generalized. Moreover, on the basis of explicit cellular decompositions generalizing the construction of Fuchs and Vainshtein, the first cohomology groups for surfaces of low genus are given.