  We introduce a new method to compute explicit formulae for various zeta
functions associated to groups and rings. The specific form of these formulae
enables us to deduce local functional equations. More precisely, we prove local
functional equations for the subring zeta functions associated to rings, the
subgroup, conjugacy and representation zeta functions of finitely generated,
torsion-free nilpotent (or $\T$-)groups, and the normal zeta functions of
$\T$-groups of class 2. In particular we solve the two problems posed in
\cite[Section 5]{duSG/06}. We deduce our theorems from a `blueprint result' on
certain $p$-adic integrals which generalises work of Denef and others on
Igusa's local zeta function. The Malcev correspondence and a Kirillov-type
theory developed by Howe are used to `linearise' the problems of counting
subgroups and representations in $\T$-groups, respectively.
