Apollonian Circle Packings and Beyond

An Apollonian circle packing is an ancient Greek construction which is made by repeatedly inscribing circles into the triangular interstices of four mutually tangent circles, via an old theorem of Apollonius of Perga (262-190 BC). We will discuss counting and equidistribution results for circles in Apollonian circle packings as well as some Diophantine questions for "integral" Apollonian packings. We will also report on some recent work on counting small tori.