Infinite circle packings like the Apollonian gasket naturally arise in many geometric and dynamical settings. For example, they can be limit sets of Kleinian groups (and more generally, boundaries of relatively hyperbolic groups) as well as homeomorphic to Julia sets of rational maps. In this talk, I will give an overview of some recent works (joint with Yusheng Luo and with Seonhee Lim and Kangrae Park) on existence and (non)rigidity of infinite circle packings, their quasiconformal geometry, and their connections with and applications to hyperbolic geometry, Teichmüller theory and Diophantine approximation.