A significant part of the legacy of Ilya Piatetski-Shapiro revolves around the method of integral representations of automorphic L-functions, a method which, in some disguise, is invoked in most occurrences of L-functions in number theory. The goal of these talks will be to describe some developments on this method since the contributions of Piatetski-Shapiro. More precisely, in the first talk I will explain a conjectural framework, and some results, by whichan affine spherical variety gives rise to automorphic distributions thatgeneralize the doubling method, and other Rankin–Selberg constructions of automorphic L-functions. This turns the theory of automorphic L-functions to a local and global problem of harmonic analysis on certain (possibly singular) spaces. In the second talk, I will discuss a further reduction to the quotient stacks associated to Jacquet's relative trace formula. In this setting, integral representations of L-functions blend with Langlands' functoriality conjecture into a broader "relative functoriality"conjecture. I will describe a new way to compare relative trace formulas, which leads to a resolution of this conjecture in rank one.