  We survey recent results on endomorphisms and especially on automorphisms of
the Cuntz algebras O_n, with a special emphasis on the structure of the Weyl
group. We discuss endomorphisms globally preserving the diagonal MASA and their
corresponding actions. In particular, we investigate those endomorphisms of O_n
which restrict to automorphisms of the diagonal. We review a combinatorial
approach to the study of permutative endomorphisms. All the presented material
is put in context with current research topics.
