Periods, equidistribution, subconvexity and non-vanishing of L-functions

In this series of lectures, we explain through examples how various periods associated to automorphic forms (Rankin/Selberg integrals, triple products, toric periods) enter and intertwine in the study of the analytic properties of L-functions; in particular, we will consider the subconvexity problem and the problem of non-vanishing at the central point.