  We consider Schr\"odinger operators $H=-\Delta+V({\mathbf x})$ in ${\mathbb
R}^d$, $d\geq2$, with quasi-periodic potentials $V({\mathbf x})$. We prove that
the absolutely continuous spectrum of a generic $H$ contains a semi-axis
$[\lambda_*,+\infty)$. We also construct a family of eigenfunctions of the
absolutely continuous spectrum; these eigenfunctions are small perturbations of
the exponentials. The proof is based on a version of the multi-scale analysis
in the momentum space with several new ideas introduced along the way.
