Using the Caffarelli–Silvestre extension, we show for a general open set Ω⊂Rn that a boundary point x0 is regular for the fractional Laplace equation (−Δ)s⁢u=0, 0<s<1, if and only if (x0,0) is regular for the extended weighted equation in a subset of Rn+1. As a consequence, we characterize regular boundary points for (−Δ)s⁢u=0 by a Wiener criterion involving a Besov capacity. A decay estimate for the solutions near regular boundary points and the Kellogg property are also obtained.