  We study nuclear embeddings for spaces of Besov and Triebel-Lizorkin type
defined on quasi-bounded domains $\Omega\subset {\mathbb R}^d$. The counterpart
for such function spaces defined on bounded domains has been considered for a
long time and the complete answer was obtained only recently. Compact
embeddings for function spaces defined on quasi-bounded domains have been
studied in detail already, also concerning their entropy and $s$-numbers. We
now prove the first and complete nuclearity result in this context. The concept
of nuclearity has been introduce by Grothendieck in 1955 already. Our second
main contribution is the generalisation of the famous Tong result (1969) which
characterises nuclear diagonal operators acting between sequence spaces of
$\ell_r$ type, $1\leq r\leq\infty$. We can now extend this to the setting of
general vector-valued sequence spaces of type $\ell_q(\beta_j \ell_p^{M_j})$
with $1\leq p,q\leq\infty$, $M_j\in {\mathbb N}_0$ and weight sequences with
$\beta_j>0$. In particular, we prove a criterion for the embedding $id_\beta :
\ell_{q_1}(\beta_j \ell_{p_1}^{M_j}) \hookrightarrow
\ell_{q_2}(\ell_{p_2}^{M_j})$ to be nuclear.
