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.