  For Denjoy--Carleman differential function classes $C^M$ where the weight
sequence $M=(M_k)$ is logarithmically convex, stable under derivations, and
non-quasianalytic of moderate growth, we prove the following: A mapping is
$C^M$ if it maps $C^M$-curves to $C^M$-curves. The category of $C^M$-mappings
is cartesian closed in the sense that $C^M(E,C^M(F,G))\cong C^M(E\x F, G)$ for
convenient vector spaces. Applications to manifolds of mappings are given: The
group of $C^M$-diffeomorphisms is a $C^M$-Lie group but not better.
