  We provide an analysis of the dynamics of isometries and semicontractions of
metric spaces. Certain subsets of the boundary at infinity play a fundamental
role and are identified completely for the standard boundaries of
CAT(0)-spaces, Gromov hyperbolic spaces, Hilbert geometries, certain
pseudoconvex domains, and partially for Thurston's boundary of Teichmueller
spaces. We present several rather general results concerning groups of
isometries, as well as the proof of other more specific new theorems, for
example concerning the existence of free nonabelian subgroups in
CAT(0)-geometry, iteration of holomorphic maps, a metric Furstenberg lemma,
random walks on groups, noncompactness of automorphism groups of convex cones,
and boundary behaviour of Kobayashi's metric.
