The Balmer spectrum of a tensor-triangulated category is the space of its prime thick tensor ideals, much like the spectrum of a ring; its open sets roughly classify Verdier localizations of the category, analogous to the way that the spectrum of a ring classifies localizations of a ring. Using some higher category theory, it is possible to equip this space with a structure sheaf and perform gluing/descent arguments as one would do in algebraic geometry. Time permitting, I will try to sketch how to rebuild a scheme from its triangulated category of perfect complexes, and show how this generalizes in our framework to provide derived algebro-geometric models for categories arising in representation theory.