Multiplicative chaos is the general name for a family of probabilistic objects, which can be thought of as the random measures obtained by taking the exponential of correlated Gaussian random variables. Multiplicative chaos turns out to be closely connected with various problems in analytic number theory, including the large values and value distribution of the Riemann zeta function in intervals on the critical line, the moments of character sums, and various model versions of these problems involving random multiplicative function. I will try to give a gentle introduction and survey of some of these issues, concentrating on the more number theoretic aspects.