The convenient setting for Denjoy-Carleman differentiable mappings of Beurling and Roumieu type

We prove in a uniform way that all Denjoy-Carleman differentiable function classes of Beurling type C-(M) and of Roumieu type C-{M}, admit a convenient setting if the weight sequence M = (M-k) is log-convex and of moderate growth: For C denoting either C-(M) or C-{M}, the category of C-mappings is cartesian closed in the sense that C(E, C(F, G))congruent to C(E x F, G) for convenient vector spaces. Applications to manifolds of mappings are given: The group of C-diffeomorphisms is a regular C-Lie group if C superset of C-omega, but not better.