Minimal submanifolds are "volume efficient shapes." Classic examples include minimal surfaces in and geodesics in a Riemannian manifold. However, despite their name, minimal submanifolds are only critical points for the volume functional, and need not be actual volume-minimizers. In fact, volume-minimizing submanifolds are quite hard to find! In this talk, we will explain how volume-minimizing submanifolds arise naturally in manifolds with exceptional geometric structures. These structures are intimately related to the normed division algebras (R, C, H, and O), and are of interest in Riemannian geometry, gauge theory and theoretical physics. Time permitting, we will mention some new examples of 4-dimensional volume-minimizing cones inside an 8-manifold modeled on the octonions.