  In this paper we give global characterisations of Gevrey-Roumieu and
Gevrey-Beurling spaces of ultradifferentiable functions on compact Lie groups
in terms of the representation theory of the group and the spectrum of the
Laplace-Beltrami operator. Furthermore, we characterise their duals, the spaces
of corresponding ultradistributions. For the latter, the proof is based on
first obtaining the characterisation of their $\alpha$-duals in the sense of
Koethe and the theory of sequence spaces. We also give the corresponding
characterisations on compact homogeneous spaces.
