In many situations in Newtonian gravity, understanding the motion of the center of mass of a system is key to understanding the general "trend" of the motion of the system. It is thus desirable to also devise a notion of center of mass with similar properties in general relativity. However, while the definition of the center of mass via the mass density is straightforward in Newtonian gravity, there is a priori no definitive corresponding notion in general relativity, let alone in the asymptotically hyperbolic setting. I will present a geometric approach to defining the center of mass of an asymptotically hyperbolic initial data set, using foliations by constant mean curvature near the asymptotically hyperbolic end of the initial data set. This approach is joint work with Cortier and Sakovich, builds upon work by Neves and Tian, and extends results in the asymptotically Euclidean case going back to Huisken and Yau.