Composition in ultradifferentiable classes

We characterize stability under composition of ultradifferentiable classes defined by weight sequences $M$, by weight functions $\omega$, and, more generally, by weight matrices $\mathfrak{M}$, and investigate continuity of composition $(g,f)\mapsto f\circ g$. In addition, we represent the Beurling space ${\mathcal{E}}^{(\omega )}$ and the Roumieu space ${\mathcal{E}}^{\{\omega \}}$ as intersection and union of spaces ${\mathcal{E}}^{(M)}$ and ${\mathcal{E}}^{\{M\}}$ for associated weight sequences, respectively.