The moments of the Riemann zeta function were introduced by Hardy and Littlewood more than 100 years ago, in an attempt to prove the Lindelöf hypothesis, which provides a strong upper bound on the size of the Riemann zeta function on the critical line. Since then, moments became central objects of study in number theory. I will give an overview of the problem of computing moments in different families of L-functions, and I will discuss some of the applications. For example, I will explain how one can extract information about the values of L-functions at special points by computing moments of the L-functions in question.