Hyperbolic 3-space does not have a well defined center of mass nor does it admit spheres with mean curvature equal to 2. Nonetheless, in many physical situations, one has to consider metrics which are perturbations of the standard hyperbolic metric. In this case, I will explain how we can define its center of mass and how the existence of spheres with constant mean curvature 2 is expected to influence the overall geometry. This is joint work with Gang Tian.