This talk is concerned with the Kolmogorov-Arnold-Moser (KAM) theorem in Gevrey classes for analytic hamiltonians, the effective stability around the corresponding KAM tori, and the semi-classical asymptotics for Schrödinger operators with exponentially small error terms. Given a real analytic Hamiltonian $H$ close to a completely integrable one and a suitable Cantor set $\Theta$ defined by a Diophantine condition, we find a family ${\Lambda }_{\omega },\omega \in \Theta$, of KAM invariant tori of $H$ with frequencies $\omega \in \Theta$ which is Gevrey smooth with...