  We show that the topology of uniform convergence on bounded sets is
compatible with the group law of the automorphism group of a large class of
spaces that are endowed with both a uniform structure and a bornology, thus
yielding numerous examples of topological groups. This encompasses many of
previously known cases, such as homeomorphism groups of locally compact spaces,
automorphism groups of first-order structures, general linear groups of normed
spaces, automorphism groups of uniform spaces, etc. We also study the
automorphism group of a topological group within this framework (generalizing
previously known results on locally compact groups). As a particular outcome of
this machinery, we show that the automorphism group of a Polish locally
Roelcke-precompact group carries a natural Polish group topology.
