Analogously to random matrices, a suitable ensemble of large random emph{tensors} has recently been proved to possess a well-defined large $N$ limit, albeit with an extra emph{universality} property. In this talk I will show that, in the context of infinite-range $p$-spin glass models, this property translates into the universality of the spin-glass transition with respect to an infinite class of perturbations of the Gaussian distribution over quenched coupling variables. I will also comment on a natural generalization of the Kosterlitz-Thouless-Jones ansatz-free solution of the spherical model to the $p$-spin spherical spin glass, and speculate on the possible applications of random tensors to short-range models as well.