  We study nuclear embeddings for spaces of Morrey type, both in its sequence
space version and as smoothness spaces of functions defined on a bounded domain
$\Omega \subset {\mathbb R}^d$. This covers, in particular, the meanwhile
well-known and completely answered situation for spaces of Besov and
Triebel-Lizorkin type defined on bounded domains which has been considered for
a long time. The complete result was obtained only recently. Compact embeddings
for function spaces of Morrey type have already been studied in detail, also
concerning their entropy and approximation numbers. We now prove the first and
complete nuclearity result in this context. The concept of nuclearity has
already been introduced by Grothendieck in 1955. Again we rely on suitable
wavelet decomposition techniques and the famous Tong result (1969) which
characterises nuclear diagonal operators acting between sequence spaces of
$\ell_r$ type, $1 \leq r \leq\infty$.
