We will start by defining random walks on Gromov-hyperbolic spaces and surveying basic results. In particular, we will talk about such notions and results as the drift (average escape rate), stationary measures, central limit theorem etc. In a second part, we will focus on the theory of large deviations in this context and give an overview of results obtained in collaboration with several co-authors: R. Aoun, A. Boulanger, P. Mathieu and A. Sisto.