  We obtain a complete classification of the continuous unitary representations
of oligomorphic permutation groups (those include the infinite permutation
group $S_\infty$, the automorphism group of the countable dense linear order,
the homeomorphism group of the Cantor space, etc.). Our main result is that all
irreducible representations of such groups are obtained by induction from
representations of finite quotients of open subgroups and moreover, every
representation is a sum of irreducibles. As an application, we prove that many
oligomorphic groups have property (T). We also show that the Gelfand--Raikov
theorem holds for topological subgroups of $S_\infty$: for all such groups,
continuous irreducible representations separate points in the group.
