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We prove in a uniform way that all Denjoy--Carleman differentiable functionclasses of Beurling type $C^{(M)}$ and of Roumieu type $C^{\{M\}}$, admit aconvenient setting if the weight sequence $M=(M_k)$ is log-convex and ofmoderate growth: For $\mathcal C$ denoting either $C^{(M)}$ or $C^{\{M\}}$, thecategory of $\mathcal C$-mappings is cartesian closed in the sense that$\mathcal C(E,\mathcal C(F,G))\cong \mathcal C(E\times F, G)$ for convenientvector spaces. Applications to manifolds of mappings are given: The group of$\mathcal C$-diffeomorphisms is a regular $\mathcal C$-Lie group if $\mathcal C\supseteq C^\omega$, but not better.

The Convenient Setting for Denjoy--Carleman Differentiable Mappings of Beurling and Roumieu Type