We prove the existence of shape resonances for Schrödinger operators of the form H(λ)=−Δ+λ2V+U, λ=n−1, in the semiclassical limit in any number of dimensions. The potential V is non-negative, vanishing at infinity as 0(|x|−α),α>0, and forms a barrier about a compact region in which V has finitely many zeros. U∈Lloc 2 is any real function which is bounded above and continuous except at a finite number of points. In addition, V and U are assumed to be dilation analytic in a neighborhood of infinity. The resonances shown to exist correspond as λ→∞ to the eigenvalues of a particle confined to the region containing the zeros of V. The width of a resonance near one of these eigenvalues λE is proved to be bounded above by c exp(−2β(λ)(ρE−ε)), for any ε>0 and where c>0 is a constant. β(λ) depends upon α and is given by λ for α>2, λ ln λ for α=2, and λ1/α+1/2 for 0<α<2. The factor ρE satisfies $$\mathop {\lim }\limits_{\lambda \to \infty } \rho _E (\lambda ) < \infty$$ , and β(λ)ρE is the leading asymptotic to the geodesic distance in the Agmon metric ds2=(λ2V+U−λE)+dx2 between the turning surfaces given by {x|λ2V+U=λE}.