Abstract: In this talk we present a weight theory in the setting of hyperbolic spaces. Our starting point is a variant of the well-known endpoint Fefferman-Stein inequality for the centered Hardy-Littlewood maximal function. Our approach is based on a combination of geometrical arguments and techniques used in the discrete setting of regular trees by Naor and Tao (2010). If time permits, we will also mention some results for $p>1$ and explain the challenges in obtaining similar results in the context of non-compact symmetric spaces in general. This talk is based on a joint work with J. Antezana.