  For a locally finite set, $A \subseteq \mathbb{R}^d$, the $k$-th Brillouin
zone of $a \in A$ is the region of points $x \in \mathbb{R}^d$ for which
$\|x-a\|$ is the $k$-th smallest among the Euclidean distances between $x$ and
the points in $A$. If $A$ is a lattice, the $k$-th Brillouin zones of the
points in $A$ are translates of each other, which tile space. Depending on the
value of $k$, they express medium- or long-range order in the set. We study
fundamental geometric and combinatorial properties of Brillouin zones, focusing
on the integer lattice and its perturbations. Our results include the stability
of a Brillouin zone under perturbations, a linear upper bound on the number of
chambers in a zone for lattices in $\mathbb{R}^2$, and the convergence of the
maximum volume of a chamber to zero for the integer lattice.
