A bounded Apollonian circle packing (ACP) is an ancient Greek construction which is made by repeatedly inscribing circles into the triangular interstices in a Descartes configuration of four mutually tangent circles. The curvatures of the circles in such a packing can be represented as coordinates of vectors in some orbit of a subgroup of O(3,1). Remarkably, if the original four circles have integer curvature, all of the circles in the packing will have integer curvature as well. In this talk we give a proof of a conjecture of Graham, Lagarias, Mallows, Wilkes, and Yan that the integers appearing as curvatures in a bounded integer ACP have positive density. We’ll also discuss some related results and make some conjectures (which are unfortunately rather out of reach at the moment).