  Apollonian circle packings arise by repeatedly filling the interstices
between four mutually tangent circles with further tangent circles. We observe
that there exist Apollonian packings which have strong integrality properties,
in which all circles in the packing have integer curvatures and rational
centers such that (curvature)$\times$(center) is an integer vector. This series
of papers explain such properties. A {\em Descartes configuration} is a set of
four mutually tangent circles with disjoint interiors. We describe the space of
all Descartes configurations using a coordinate system $\sM_\DD$ consisting of
those $4 \times 4$ real matrices $\bW$ with $\bW^T \bQ_{D} \bW = \bQ_{W}$ where
$\bQ_D$ is the matrix of the Descartes quadratic form $Q_D= x_1^2 + x_2^2+
x_3^2 + x_4^2 -{1/2}(x_1 +x_2 +x_3 + x_4)^2$ and $\bQ_W$ of the quadratic form
$Q_W = -8x_1x_2 + 2x_3^2 + 2x_4^2$. There are natural group actions on the
parameter space $\sM_\DD$. We observe that the Descartes configurations in each
Apollonian packing form an orbit under a certain finitely generated discrete
group, the {\em Apollonian group}. This group consists of $4 \times 4$ integer
matrices, and its integrality properties lead to the integrality properties
observed in some Apollonian circle packings. We introduce two more related
finitely generated groups, the dual Apollonian group and the super-Apollonian
group, which have nice geometrically interpretations. We show these groups are
hyperbolic Coxeter groups.
