The asymptotic entropy of a random walk on a countable group is a non-negative number that determines the existence of non-constant bounded harmonic functions on the group. A natural question to ask is whether the asymptotic entropy, seen as a function of the step distribution of the random walk, is continuous. In this talk, I will explain two recent results on the continuity of asymptotic entropy: one for groups whose Poisson boundaries can be identified with a compact metric space carrying a unique stationary measure, and another for wreath products A ≀ Z^d, where A is a countable group and d ≥ 3.