**Abstract** In this talk we discuss the Grothendieck-Lidskii formula in quaternionic Hilbert spaces, along with the particularities of the problem, which include, among others, the definition of an appropriate trace that differs from the usual one.

We will also briefly discuss $r$-nuclear operators in quaternionic Banach (or, more generally, locally convex) spaces $X$, i.e., operators of the form $$ \sum_{k=1}^\infty \mu_k (x_k\otimes x_k’),\qquad x_k\in X,\ x_k’\in X’, $$ where ${\mu_k}\in \ell^r$, and the associated Grothendieck-Lidskii’s formula for the case $r\leq 2/3$. In order to establish these results, the use of the aforementioned traces (and their invariance with respect to the basis choices) is essential.

To conclude, we briefly discuss how the (seemingly ad hoc) introduced trace actually arise as the canonical form in tensor products of quaternionic vector spaces. This is a joint work with P. Cerejeiras, F. Colombo, U. Kähler, and I. Sabadini.