Let $K$ be a smooth or polyhedral bounded subset in the hyperbolic space $H^3$. Then $K$ is uniquely determined by the induced metric on its boundary or, dually, by its third fundamental form (or dual metric). When one considers more generally a hyperbolic manifold with convex boundary, similar statements are known, while others remain conjectural, including in what is perhaps the simplest case, the convex core of a quasifuchsian manifold. We will describe recent results on this question, including by Abderrahim Mesbah and Diptaishik Choudhury, as well as recent joint work with with Qiyu Chen.