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.