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.