This talk focuses on asymptotic properties of geometric branching processes on hyperbolic spaces and manifolds. (In certain aspects, processes on hyperbolic spaces are simpler than on Euclidean spaces.) The first paper in this direction was by Lalley and Sellke (1997) and dealt with a homogenous branching diffusion on a hyperbolic (Lobachevsky) plane). Afterwards, Karpelevich, Pechersky and Suhov (1998) extended it to general homogeneous branching processes on hyperbolic spaces of any dimension. Later on, Kelbert and Suhov (2006, 2007) proceeded to include non-homogeneous branching processes. One of the main questions here is to calculate the Hausdorff dimension of the limiting set on the absolute. I will not assume any preliminary knowledge of hyperbolic geometry.