  This survey is concerned with random walks on mapping class groups. We
illustrate how the actions of mapping class groups on Teichm\"uller spaces or
curve complexes reveal the nature of random walks, and vice versa. Our emphasis
is on the analogues of classical theorems, including laws of large numbers and
central limit theorems, and the properties of harmonic measures, under optimal
moment conditions. We also explain the geometric analogy between Gromov
hyperbolic spaces and Teichm\"uller spaces that has been used to copy the
properties of random walks from one to the other.
