The study of moments of various families of automorphic L-functions has for a long time been an important subject in number theory. In the absence of the Riemann Hypothesis, or of the Grand Riemann Hypothesis referring to general L-functions, suitable results on moments often served as a substitute. In this talk, I will discuss some classical moment problems and their connexion with harmonic analysis. As an application, I will outline the main ideas of establishing a subconvexity result for GL_2 automorphic L-functions over arbitrary number fields. This is based on joint work with Paul Garrett and Dorian Goldfeld.