The Denjoy-Schwartz theorem for surfaces says that any flow topologically equivalent to a \(C^ 2\) flow has only trivial minimal sets. The author proves the converse. In fact, he shows that minimal sets are trivial if and only if the flow is topologically equivalent to a \(C^{\infty}\) flow. He also gives a structure theorem and existence theorem for flows on surfaces with nontrivial recurrent orbits, via interval exchange transformations.