  In the last years the biregular automorphisms of the 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 the Kapranov's and Keel's
realizations of $\bar{M}_{0,n}$, and to the degenerations of Hassett's spaces
obtained by allowing zero weights.
