  We study almost periodic pseudodifferential operators acting on almost
periodic functions $G_{\rm ap}^s(\rr d)$ of Gevrey regularity index $s \geq 1$.
We prove that almost periodic operators with symbols of H\"ormander type
$S_{\rho,\delta}^m$ satisfying an $s$-Gevrey condition are continuous on
$G_{\rm ap}^s(\rr d)$ provided $0 < \rho \leq 1$, $\delta=0$ and $s \rho \geq
1$. A calculus is developed for symbols and operators using a notion of
regularizing operator adapted to almost periodic Gevrey functions and its
duality. We apply the results to show a regularity result in this context for a
class of hypoelliptic operators.
