the gross-prasad conjecture says that the period integral of a cuspidal representation of so(n+1) x so(n) over the diagonal so(n) is nonzero if and only if the central critical value of a certain l-function does not vanish. there is a refinement of the conjecture, due to ichino-ikeda, which gives a formula for the central l-value in terms of the period integral. i will explain this refined conjecture, and discuss some joint work with ichino in the case n=4. the case n=2 is a classic theorem of waldspurger and the case n=3 is a recent theorem of ichino.