We introduce a generalization of commutative rings called bands, along with the corresponding geometric theory of band schemes. Among other things, band schemes provide a new viewpoint on tropical geometry and Berkovich analytifications, and they enable a partial explanation for phenomena observed by Jacques Tits concerning algebraic groups over the "field of one element". Band schemes also furnish a natural algebro-geometric setting for studying matroid theory; in particular, they allow us to construct a moduli space of matroids, and they provide new tools for studying realization spaces of matroids. If time permits, we will discuss applications to a generalization of Laurent Lafforgue's theorem on realization spaces of rigid matroids. This is joint work with Oliver Lorscheid and Tong Jin.