Volume functions and boundary data of 3-dimensional hyperbolic manifolds

We review recent progress on two closely related sets of questions concerning convex co-compact hyperbolic manifolds, or convex domains in those manifolds, such as their convex core. The first set of questions is to what extent the hyperbolic metric on such a manifold is uniquely determined by either of two possible geometric data on their boundary. The second aspect is the volume'' associated to such a manifold, such as the renormalized volume of a convex co-compact hyperbolic manifold. The relation between the two is provided by the first variation of the volume functions, which involves the two kinds of boundary data as conjugate’’ variables. While progress has recently been made on some questions, others remain open. New connections have recently emerged, with physics (and in particular the AdS/CFT correspondence) as well as with probability theory (the Loewner energy).