  The Zak transform on $\mathbb{R}^d$ is an important tool in condensed matter
physics, signal processing, time-frequency analysis, and harmonic analysis in
general. This article introduces a generalization of the Zak transform to a
class of locally compact $G$-spaces, where $G$ is either a locally compact
abelian or a second countable unimodular type I group. This framework unifies
previously proposed generalizations of the Zak transform. It is shown that the
Zak transform has invariance properties analog to the classic case and is a
Hilbert space isomorphism between the space of $L^2$-functions and a direct
integral of Hilbert spaces that is explicitly determined via a Weil formula for
$G$-spaces and a Poisson summation formula for compact subgroups. Some
applications in physics are outlined.
