  We study topological properties of conjugacy classes in Polish groups, with
emphasis on automorphism groups of homogeneous countable structures. We first
consider the existence of dense conjugacy classes (the topological Rokhlin
property). We then characterize when an automorphism group admits a comeager
conjugacy class (answering a question of Truss) and apply this to show that the
homeomorphism group of the Cantor space has a comeager conjugacy class
(answering a question of Akin-Hurley-Kennedy). Finally, we study Polish groups
that admit comeager conjugacy classes in any dimension (in which case the
groups are said to admit ample generics). We show that Polish groups with ample
generics have the small index property (generalizing results of
Hodges-Hodkinson-Lascar-Shelah) and arbitrary homomorphisms from such groups
into separable groups are automatically continuous. Moreover, in the case of
oligomorphic permutation groups, they have uncountable cofinality and the
Bergman property. These results in particular apply to automorphism groups of
many $\omega$-stable, $\aleph_0$-categorical structures and of the random
graph. In this connection, we also show that the infinite symmetric group
$S_\infty$ has a unique non-trivial separable group topology. For several
interesting groups we also establish Serre's properties (FH) and (FA).
