  We study Lie group structures on groups of the form C^\infty(M,K)}, where M
is a non-compact smooth manifold and K is a, possibly infinite-dimensional, Lie
group. First we prove that there is at most one Lie group structure with Lie
algebra C^\infty(M,k) for which the evaluation map is smooth. We then prove the
existence of such a structure if the universal cover of K is diffeomorphic to a
locally convex space and if the image of the left logarithmic derivative in
\Omega^1(M,k) is a smooth submanifold, the latter being the case in particular
if M is one-dimensional. We also obtain analogs of these results for the group
O(M,K) of holomorphic maps on a complex manifold with values in a complex Lie
group. We show that there exists a natural Lie group structure on O(M,K) if K
is Banach and M is a non-compact complex curve with finitely generated
fundamental group.
