  Tree-graded spaces are generalizations of R-trees. They appear as asymptotic
cones of groups (when the cones have cut points). Since many questions about
endomorphisms and automorphisms of groups, solving equations over groups,
studying embeddings of a group into another group, etc. lead to actions of
groups on the asymptotic cones, it is natural to consider actions of groups on
tree-graded spaces. We develop a theory of such actions which generalizes the
well known theory of groups acting on R-trees. As applications of our theory,
we describe, in particular, relatively hyperbolic groups with infinite groups
of outer automorphisms, and co-Hopfian relatively hyperbolic groups.
