  We study and classify proper $q$-colorings of the $\mathbb Z^d$ lattice,
identifying three regimes where different combinatorial behavior holds: (1)
When $q\le d+1$, there exist frozen colorings, that is, proper $q$-colorings of
$\mathbb Z^d$ which cannot be modified on any finite subset. (2) We prove a
strong list-coloring property which implies that, when $q\ge d+2$, any proper
$q$-coloring of the boundary of a box of side length $n \ge d+2$ can be
extended to a proper $q$-coloring of the entire box. (3) When $q\geq 2d+1$, the
latter holds for any $n \ge 1$. Consequently, we classify the space of proper
$q$-colorings of the $\mathbb Z^d$ lattice by their mixing properties.
