  We characterize the nuclearity of the Beurling-Bj\"{o}rck spaces
$\mathcal{S}^{(\omega)}_{(\eta)}(\mathbb{R}^d)$ and
$\mathcal{S}^{\{\omega\}}_{\{\eta\}}(\mathbb{R}^d)$ in terms of the defining
weight functions $\omega$ and $\eta$.
