Contents

We present a conceptual and uniform interpretation of the methods of integral representations of L-functions (period integrals, Rankin-Selberg integrals). This leads to: (i) a way to classify such integrals, based on the classification of certain embeddings of spherical varieties (whenever the latter is available), (ii) a conjecture which would imply a vast generaliza-tion of the method, and (iii) an explanation of the phenomenon of “weight factors ” in a relative trace formula. We also prove results of independent interest, such as the generalized Cartan decomposition for spherical vari-eties of split groups over p-adic fields (following an argument of Gaitsgory and Nadler)