On the automorphisms of moduli spaces of curves

In the last years the biregular automorphisms of Deligne–Mumford’s and Hassett’s compactifications of the moduli space of n-pointed genus g smooth curves have been extensively studied by A. Bruno and the authors. In this paper we give a survey of these recent results and extend our techniques to some moduli spaces appearing as intermediate steps of Kapranov’s and Keel’s realizations of M<inf>0;n</inf>, and to the degenerations of Hassett’s spaces obtained by allowing zero weights.