  We introduce and study a number of new spaces of ultradifferentiable
functions and ultradistributions and we apply our results to the study of the
convolution of ultradistributions. The spaces of convolutors
$\mathcal{O}'^{\ast}_{C}(\mathbb{R}^{d})$ for tempered ultradistributions are
analyzed via the duality with respect to the test function spaces
$\mathcal{O}^{\ast}_{C}(\mathbb{R}^{d})$, introduced in this article. We also
study ultradistribution spaces associated to translation-invariant Banach
spaces of tempered ultradistributions and use their properties to provide a
full characterization of the general convolution of Roumieu ultradistributions
via the space of integrable ultradistributions. We show that the convolution of
two Roumieu ultradistributions $T,S\in \DD'^{\{M_p\}}\left(\RR^d\right)$ exists
if and only if
$\left(\varphi*\check{S}\right)T\in\DD'^{\{M_p\}}_{L^1}\left(\RR^d\right)$ for
every $\varphi\in\DD^{\{M_p\}}\left(\RR^d\right)$.
