  Recently, much interest was devoted to the Urysohn universal metric space U
and its isometries; this paper is a contribution to this field of research. In
particular, we study some properties of isometries of U, and prove the
following result, which answers a question of Urysohn: if X is a subset of U is
such that, for any X' which is isometric to X there exists an isometry of U
which maps X to X', then X is compact (the converse was already well-known). We
also answer in the negative a question of Clemens, proving that any Polish
metric space is isometric to the set of fixed points of some isometry of U.
