Following Jacquet, Lapid and Rogawski, we define a regularized period of an automorphic form on GL(n+1) x GL(n) along the diagonal subgroup GL(n) and express it in terms of the Rankin-Selberg integral of Jacquet, Piatetski-Shapiro and Shalika. This extends the theory of the Rankin-Selberg integrals to all automorphic forms on GL(n+1) x GL(n). We also discuss the case of unitary groups and give some application to the global Gan-Gross-Prasad conjecture in the endoscopic case. This is a joint work with Shunsuke Yamana.