  We study the asymptotic limiting function $W({G},q) = \lim_{n \to
\infty}P(G,q)^{1/n}$, where $P(G,q)$ is the chromatic polynomial for a graph
$G$ with $n$ vertices. We first discuss a subtlety in the definition of
$W({G},q)$ resulting from the fact that at certain special points $q_s$, the
following limits do not commute: $\lim_{n \to \infty} \lim_{q \to q_s}
P(G,q)^{1/n} \ne \lim_{q \to q_s} \lim_{n \to \infty} P(G,q)^{1/n}$. We then
present exact calculations of $W({G},q)$ and determine the corresponding
analytic structure in the complex $q$ plane for a number of families of graphs
${G}$, including circuits, wheels, biwheels, bipyramids, and (cyclic and
twisted) ladders. We study the zeros of the corresponding chromatic polynomials
and prove a theorem that for certain families of graphs, all but a finite
number of the zeros lie exactly on a unit circle, whose position depends on the
family. Using the connection of $P(G,q)$ with the zero-temperature Potts
antiferromagnet, we derive a theorem concerning the maximal finite real point
of non-analyticity in $W({G},q)$, denoted $q_c$ and apply this theorem to
deduce that $q_c(sq)=3$ and $q_c(hc) = (3+\sqrt{5})/2$ for the square and
honeycomb lattices. Finally, numerical calculations of $W(hc,q)$ and $W(sq,q)$
are presented and compared with series expansions and bounds.
