  In this paper, we define and study Gevrey spaces associated with a
H\"ormander family of (globally defined) vector fields and its corresponding
sub-Laplacian. We show some natural relations between the various Gevrey spaces
in this setting on general manifolds, and more particular properties on Lie
groups with polynomial growth of the volume. In the case of the Heisenberg
group and of $SU(2)$, we show that all our descriptions coincide.
