We will review the notion of mass for asymptotically hyperbolic manifolds, and discuss the respective positive mass conjecture and Penrose conjecture. We will also address the nearly equality case of the positive mass theorem: if the mass is small, can we say that the metric is close to the hyperbolic metric in some sense? A part of the new results to be presented comes from joint work with Mattias Dahl and Romain Gicquaud.