Apollonian circle packings are infinite packings of circles, constructed recursively from an initial configuration of four mutually touching circles by adding circles externally tangent to triples of such circles. Configurations of four mutually touching circles were studied by Descartes in 1643. If the initial four circles have integer curvatures, so do all the circles in the packing. If in addition the circles have rational centers so do all the circles in the packing. Why? This talk describes results in geometry, group theory and number theory arising from such packings. (This is joint work with Ron Graham, Colin Mallows, Allan Wilks, and Catherine Yau.)