In this note we sketch our computation of the group A 2 of cobordism classes of orientation-preserving diffeomorphisms of closed, oriented surfaces. See Sect. 2 for precise definitions. This computation was made independently and a little earlier by Bonahon [2]. Our development follows in part the same lines, extending approaches to related questions by Scharleman [13] and Johannson and Johnson [10], who were the first to apply the characteristic manifold theory of Johannson [8] and Jaco and Shalen [7] to questions about surface diffeomorphisms. Bonahon's approach depends crucially on the Mostow Rigidity Theorem and the unpublished and as yet inaccessible Hyperbolization Theorem of Thurston. In contrast, we get by with the G-Signature Theorem (in dimension two, a classical formula) and some number theory. For this reason we have belatedly decided to publish our own approach. Our main contribution is a proof, independent of deep three-dimensional geometry, that an orientation-preserving periodic surface diffeomorphism which bounds as a diffeomorphism also bounds periodically. Cf. I-2 ; Proposition B]. The proof is given in Sect. 3. In Sect. 2 we develop the proof of the main calculation, modulo the analysis of periodic diffeomorphisms deferred to Sect. 3.