The Poisson boundary is a measure theoretic object that captures the space of distinct asymptotic trajectories that a random walk can take. A fundamental class of examples of groups for which one should study the Poisson boundary dating back to a fundamental paper of Kaimanovich and Vershik are lamplighter groups (or more generally wreath products). I will discuss some recent work classifying the Poisson boundary of the lamplighter groups and wreath products for a wide class of measures (including, in particular, all finite first moment measures). This is joint work with Eduardo Silva.