  This is a short survey on idempotent states on locally compact groups and
locally compact quantum groups. The central topic is the relationship between
idempotent states, subgroups and invariant C*-subalgebras. We concentrate on
recent results on locally compact quantum groups, but begin with the classical
notion of idempotent probability measure. We also consider the `intermediate'
case of idempotent states in the Fourier--Stieltjes algebra: this is the dual
case of idempotent probability measures and so an instance of idempotent states
on a locally compact quantum group.
