Packings of hyperbolic surfaces

It is well known in low-dimensional topology that 'Boroczky's Theorem', which bounds the local density of ball packings of hyperbolic space, also yields sharp density bounds for such packings of hyperbolic manifolds in certain circumstances.  Using this fact, in 1996 C. Bavard characterized the 'extremal' closed hyperbolic surfaces: those which admit an embedded disk of largest possible radius.  I will discuss the characterization of non-compact extremal surfaces of finite volume, which required new ideas, and I'll report what I know (not much) about the case of several-disk packings of hyperbolic surfaces.