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In this paper we prove the Pohozaev identity for the semilinear Dirichlet problem $${(-\Delta)^s u =f(u)}$$ in $${\Omega, u\equiv0}$$ in $${{\mathbb R}^n\backslash\Omega}$$ . Here, $${s\in(0,1)}$$ , (−Δ) is the fractional Laplacian in $${\mathbb{R}^n}$$ , and Ω is a bounded C domain. To establish the identity we use, among other things, that if u is a bounded solution then $${u/\delta^s|_{\Omega}}$$ is C up to the boundary ∂Ω, where δ( x) = dist( x, ∂Ω). In the fractional Pohozaev identity, the function $${u/\delta^s|_{\partial\Omega}}$$ plays the role that ∂u/ ∂ν plays in the classical one. Surprisingly, from a nonlocal problem we obtain an identity with a boundary term (an integral over ∂Ω) which is completely local. As an application of our identity, we deduce the nonexistence of nontrivial solutions in star-shaped domains for supercritical nonlinearities. [ABSTRACT FROM AUTHOR]