For a compact manifold M, denote by Diff(M) the topological group of diffeomorphisms of M. We survey on some old and new results concerning the homotopy groups \pi_* Diff(M) for high-dimensional spin manifolds M, along with their action on \pi_* R^+(M), where R^+(M) denotes the space of Riemannian metrics of positive scalar curvature on M. (Joint work with D. Crowley, B. Hanke and Th. Schick.)