These notes are based on a talk given at the Institut de Mathématiques Elie Cartan de Nancy in June 2006. Their purpose is to introduce the reader to some links between two fields of mathematics : analytic number theory and random matrices. After some historical overview of these connections, we expose a conjecture about the moments of the Riemann zeta function, formulated by Keating and Snaith. Last, we give some probabilistic interpretations of their corresponding results about unitary random matrices.