The Poisson boundary is a measure-theoretic object attached to a group equipped with a probability measure, and is closely related to the notion of harmonic function on the group. In many cases, the group is also endowed with a topological boundary arising from its geometric structure, and a recurring research theme is to identify the Poisson boundary with the topological boundary. For instance, for lattices in hyperbolic space, it is natural to ask if the (visual, or Gromov-)boundary of the hyperbolic space is a model for the Poisson boundary. In this talk, we solve the identification problem for the Poisson boundary of a random walk with finite entropy on a hyperbolic group and on a discrete subgroup of a semisimple Lie group. The main technical novelty is that we do not require any moment assumption on the measure. Joint with K. Chawla, B. Forghani, and J. Frisch.