Let $\Omega_1$ and $\Omega_2$ be non empty open subsets of $\mathbb R^r$ and $\mathbb R^s$ respectively
and let $\omega_1$ and $\omega_2$ be weights.
We introduce the spaces of ultradifferentiable functions $\mathcal{E}_{(\omega_1,\omega_2)}(\Omega_1 \times \Omega_2)$,
$\mathcal{D}_{(\omega_1,\omega_2)}(\Omega_1 \times \Omega_2)$,
$\mathcal{E}_{\{\omega_1,\omega_2\}}(\Omega_1 \times \Omega_2)$
and $\mathcal{D}_{\{\omega_1,\omega_2\}}(\Omega_1 \times \Omega_2)$,
study their locally convex properties, examine
the structure of their elements and also consider their links with the tensor products
$\mathcal{E}_{*}(\Omega_1) \otimes \mathcal{E}_{*}(\Omega_2)$
and $\mathcal{D}_{*}(\Omega_1) \otimes \mathcal{D}_{*}(\Omega_2)$
endowed with the $\varepsilon$-, $\pi$- or $i$-topologies.
This leads to kernel theorems.