  We prove new kernel theorems for a general class of Beurling-Bj\"orck type
spaces. In particular, our results cover the classical Beurling-Bj\"orck spaces
$\mathcal{S}^{(\omega)}_{(\eta)}$ and $\mathcal{S}^{\{\omega\}}_{\{\eta\}}$
defined via weight functions $\omega$ and $\eta$.
