Abstract: In a first part, we will discuss large deviation phenomena and existence of large deviation principle for the displacement of random walks on hyperbolic spaces after having introduced related notions from probability theory. In a second part, we will focus on finite time bounds and concentrations for deviations of the displacement off the drift and mention an application to finite-time probabilistic Tits? alternative. Based on joint works with Adrien Boulanger, Pierre Mathieu and Alessandro Sisto (part one above), and with Richard Aoun (part two above).