Abstract: This talk will provide an overview of the renormalized volume
coefficients and associated renormalized volume functionals in conformal
geometry. These are Riemannian invariants constructed from the volume
expansion of a Poincar\'e-Einstein metric with a prescribed conformal
infinity. They arose in the context of the AdS/CFT correspondence in
physics. They have some surprising properties, which, among other things,
suggests a natural variant of the so-called $\sigma_k$-Yamabe problem, a
fully nonlinear generalization of the Yamabe problem which has been the
focus of much attention during the last decade.