Abstract: The Poisson Boundary of a random walk on a group is a measure space that corresponds to the space of different asymptotic trajectories that the random walk might take. Given a group $G$ and a probability measure $\mu$ on $G$ the Poisson boundary is trivial (i.e. has no non-trivial events) if and only if $G$ supports a bounded $mu$-harmonic function. In this talk I will give an introduction to the notion of the poisson boundary and discuss a recent characterization of exactly which countable groups $G$ have a trivial Poisson Boundary for every measure $\mu$ (the so called Choquet-Deny groups). This characterization will immediately imply that, for finitely generated groups, these groups are exactly those of polynomial growth. This answers a question of Kaimanovich and Vershik. Surprisingly, the proof does not rely at all on growth estimates for a group, and instead relies on the algebraic infinite conjugacy class property. This is joint work with Yair Hartman, Omer Tamuz, and Pooya Vahidi Ferdowski.