  Apollonian circle packings arise by repeatedly filling the interstices
between mutually tangent circles with further tangent circles. It is possible
for every circle in such a packing to have integer radius of curvature, and we
call such a packing an {\em integral Apollonian circle packing.} This paper
studies number-theoretic properties of the set of integer curvatures appearing
in such packings. Each Descartes quadruple of four tangent circles in the
packing gives an integer solution to the {\em Descartes equation}, which
relates the radii of curvature of four mutually tangent circles: $2 (x^2 + y^2
+ z^2 + w^2) - (x + y + z + w)^2 = 0.$ Each integral Apollonian circle packing
is classified by a certain {\em root quadruple} of integers that satisfies the
Descartes equation, and that corresponds to a particular quadruple of circles
appearing in the packing. We determine asymptotics for the number of root
quadruples of size below $T$. We study which integers occur in a given integer
packing, and determine congruence restrictions which sometimes apply. Finally,
we present evidence suggesting that the set of integer radii of curvatures that
appear in an integral Apollonian circle packing has positive density, and in
fact represents all sufficiently large integers not excluded by congruence
conditions. In a series of companion papersr ``Apollonian Circle Packings:
Geometry and Group Theory,'' we investigate a variety of group-theoretic
properties of these configurations, as well as various extensions to higher
dimensions and other spaces, such as hyperbolic space.
