In the 1990s, Besson-Courtois-Gallot constructed a new type of geometric map between Riemannian manifolds when the target is negatively curved. This map has been extended and used in several different settings. In 2 or 3 lectures I will assume away as many technical points as possible and present the heart of the map's construction. At the nut of it lies a remarkable and mysterious computation which still has no conceptual proof.