  We study nuclear embeddings for weighted spaces of Besov and Triebel-Lizorkin
type where the weight belongs to some Muckenhoupt class and is essentially of
polynomial type. Here we can extend our previous results [17,19] where we
studied the compactness of corresponding embeddings. The concept of nuclearity
goes back to Grothendieck who defined it in [14]. Recently there is a refreshed
interest to study such questions [5-8,49]. This led us to the investigation in
the weighted setting. We obtain complete characterisations for the nuclearity
of the corresponding embedding. Essential tools are a discretisation in terms
of wavelet bases, operator ideal techniques, as well as a very useful result of
Tong [43] about the nuclearity of diagonal operators acting in $\ell_p$ spaces.
In that way we can further contribute to the characterisation of nuclear
embeddings on domains obtained in [5,33,34,49].
