A period of an automorphic form on a reductive group $G$ over anumber fieldis defined by its integral over a subgroup $H$ of $G$. Suchperiods are often related to special values of automorphic $L$-functions.In this talk, we present a conjecture in the case of special orthogonalgroups, which can be regarded as a refinement of the global Gross-Prasadconjecture about the restriction of automorphic representations of $SO(n+1$) to $SO(n)$. If time permits, we also discuss a relation of ourconjecture to Arthur's conjecture on the multiplicity of representationsin the space of automorphic forms. This is a joint work with Tamotsu Ikeda.