We will consider random walks on spaces of isometries of hyperbolic spaces, trying to understand their asymptotic properties. If the space has negative curvature, then it has a non-trivial topological boundary. We will be interested in comparing the boundary theory of the random walk with the topological notion.

1) Boundary convergence: We will prove the theorem of Furstenberg which states that a random walk on SL_2(R) converges to the boundary of the hyperbolic plane almost surely.

2) The Poisson-Furstenberg boundary: We will discuss how to attach a general notion of boundary to any random walk on a group, and what are the criteria which allow one to identify this general measurable boundary with a topological boundary which is given by the geometry.