  A topological group G is defined to have property (OB) if any G-action by
isometries on a metric space, which is separately continuous, has bounded
orbits. We study this topological analogue of the socalled Bergman property in
the context of Polish groups, where we show it to have several interesting
reformulations and consequences. We subsequently apply the results obtained in
order to verify property (OB) for a number of groups of isometries and
homeomorphism groups of compact metric spaces. We also give a proof that the
isometry group of the rational Urysohn metric space of diameter 1 is Bergman.
