On a basic level, tensor triangular geometry (tt-geometry) associates to a (symmetric) tensor triangulated category a geometric invariant, its Zariski spectrum, roughly by treating the category as a commutative ring. On a slightly higher level, the spectrum is the universal object which captures the support theory on the category, shadowing sheaf supports in algebraic geometry. Either way, given a tensor triangulated category $T$, calculating its spectrum $\operatorname{Spec} T$ gives an important global information about the category but also appears to be a highly nontrivial problem. I’ll mention some known calculations which exploit the richness of the endomorphism ring of the unit object, $\operatorname{End}^*_T(1)$, going back to the work of Quillen, but also highlight some emerging new (and old) cases where $\operatorname{End}^*_T(1)$ is rather small and cannot possibly capture the entire spectrum of $T$.