The relationship between periods of automorphic forms and special values of L-functions is a central motivation of the relative Langlands program. In this talk, I discuss my joint work with Jingwei Xiao and Wei Zhang, where we formulate a global conjecture relating periods associated to symmetric spaces of unitary groups to central values of standard L-functions on linear groups, generalizing a theorem of Waldspurger for GL(2). To attack this conjecture, we introduce a new family of relative trace formulas built from the Bump-Friedberg integral, which we successfully compare under certain local assumptions to prove cases of the conjecture. A new feature of this comparison is the presence of relative endoscopy: we need to establish an endoscopic comparison of relative trace formulas to prove our global results. All of this relies on establishing several new local results.