Quasifuchsian hyperbolic manifolds have infinite volume, but they
have a well-defined "renormalized" volume, closely related to the
Liouville functional. It has interesting "analytic" properties,
in particular it provides a Kähler potential for the Weil-Petersson
metric of the boundary at infinity, but also interesting "coarse"
properties since it differs by at most additive constants from the
volume of the convex core, and therefore from the Weil-Petersson
distance between the conformal structures at infinity. We will survey
the construction and main properties of this renormalized volume,
as well as some recent applications.