Many of the functions of greatest interest to analytic number theorists have a property called multiplicativity. Examples include the Möbius function, which is tied up with the Riemann Hypothesis; and Dirichlet characters, which are tied up with the distribution of sequences in arithmetic progressions. To guess how these functions might behave, it turns out to be fruitful to study probabilistic models. Recently, connections have been found with quite subtle probabilistic issues such as branching random walk and multiplicative chaos. It also turns out that one can (sometimes) use the probabilistic models not only to guess, but also to prove, results about the deterministic multiplicative functions that one started with. I will try to give a gentle introduction and overview of some of these issues.