Given a random walk on a group of isometries of hyperbolic (or other symmetric) space, one can consider its hitting measure, i.e. the probability that the walk converges to a given subset of the boundary. It has been discussed for several decades, starting with the work of Furstenberg in the late 60's, whether the hitting measure for a random walk can lie in the same measure class as a "nice" geometric measure, e.g. the Lebesgue measure on the boundary. It is a long-standing conjecture, formalized by Kaimanovich-Le Prince, that, if the group of isometries is discrete and the random walk is finitely supported, the hitting measure is always singular with respect to Lebesgue. We will explore this problem in various contexts, and discuss the state of the art and recent progress, based on joint works with N. Bogachev, P. Kosenko, H. Lee, and W. van Limbeek.