  We consider the reducibility problem of cocycles $(\alpha,A)$ on $\T^d\times
U(n)$ in Gevrey classes, where $ \alpha$ is a Diophantine vector. We prove
that, if a Gevrey cocycle is conjugated to a constant cocycle $(\alpha,C)$ by a
suitable measurable conjugacy $(0,B)$, then for almost all $C$ it can be
conjugated to $(\alpha,C)$ in the same Gevrey class, provided that $A$ is
sufficiently close to a constant. If $B$ is continuous we obtain it is Gevrey
smooth. We consider as well the global problem of reducibility in Gevrey
classes when $d=1$.
