Summary: We introduce an equivalence relation, called stable equivalence, on knot diagrams and closed generically immersed curves on surfaces. We give bijections between the set of abstract knots, the set of virtual knots, and the set of stable equivalence classes of knot diagrams on surfaces. Using these bijections, we define concordance and link homology for vitual links. As an application, it is shown that Kauffman’s example of a virtual knot diagram is not equivalent to a classical knot diagram.