Let $V_n\subset V_{n+1}$ be orthogonal spaces of dimensions $n$ and $n+1$ over a number field $F$, and let $G_n\subset G_{n+1}$ be the associated special orthogonal groups. Let $\pi_n$ and $\pi_{n+1}$ be irreducible, cuspidal, tempered, automorphic representations of $G_n(\mathbb{A}_F)$ and $G_{n+1}(\mathbb{A}_F)$. In the early 1990s, Gross and D. Prasad conjectured that a certain period integral attached to $\pi_n$ and $\pi_{n+1}$ is non-zero if and only if a certain automorphic $L$-function is non-zero at $s=1/2$. Recently, A. Ichino and T. Ikeda have proposed a refinement of this conjecture; they give an explicit formula relating the period integral to the $L$-value. In this talk, we state a similar conjecture for unitary groups, as well as sketch the proof of the first case.