A closed subset E of a Riemann surface R is called a set of holomorphic (resp. meromorphic) approximation if every function continuous on E and holomorphic on Int(E) can be approximated uniformly on E by functions holomorphic (resp. meromorphic) on (all of) R. When R is the complex plane, a complete characterization of the sets of approximation is known. For arbitrary noncompact Riemann surfaces, and E noncompact, the problem is still wide open.