  We give a survey of Denef's rationality theorem on $p$-adic integrals, its
uniform in $p$ versions, the relevant model theory, and a number of
applications to counting subgroups of finitely generated nilpotent groups and
conjugacy classes in congruence quotients of Chevalley groups over rings of
integers of local fields. We then state results on analytic properties of Euler
products of such $p$-adic integrals over all $p$, and an application to
counting conjugacy classes in congruence quotients of certain algebraic groups
over the rationals. We then briefly discuss zeta functions arising from
definable equivalence relations and $p$-adic elimination of imginaries, which
have applications to counting representations of groups.
