Abstract: Metric hyperbolic spaces were originally introduced by Gromov in the 80’s with the goal of extending the known notions and results about negative curvature in two directions: group theory and infinite dimension symmetric spaces. In recent years, a lot of work as been developed surrounding these spaces, namely in what concerns de dynamics of geometric actions. In this talk we do a comprehensive introduction on hyperbolic spaces presenting illustrative examples. We then proceed to introduce the notion of G-cocycle and its drift, for G some group acting by isometries on the hyperbolic space. We finish the presentation with new results concerning the drift, namely we prove Avalanche Principle type results. An important example at this point will be the usual hyperbolic plane where the notions of drift and Lyapunov exponent coincide. This talk is heavily inspired by recent works on the matter [1, 2] where the authors deeply explore the drift in the context of random walks. We shall focus these topics on a second part of the seminar. References [1] Adrien Boulanger, Pierre Mathieu, Cagri Sert, and Alessandro Sisto. Large deviations for random walks on hyperbolic spaces. arXiv preprint arXiv:2008.02709, 2020. [2] Joseph Maher and Giulio Tiozzo. Random walks on weakly hyperbolic groups. Journal f ̈ur die reine und angewandte Mathematik, 2018(742):187–239, 2018.