Abstract: In the quantum theory of solids, a metal is distinguished from an insulator by having a Fermi surface - a surface (in momentum space) where "the drama of the life of the electron is played out". Understanding how electrons behave on the Fermi surface is crucial to understanding the basic properties of metals, such as their lustrous appearance, and their ability to conduct heat and electricity. Solid-state physicists have developed experimental techniques to trace out the geometrical shape of the Fermi surface – to an accuracy of one in a thousand. However, certain emergent properties of metals are not determined by Fermi-surface geometry, but instead depend on the quantum-mechanical wave function of electrons on the Fermi surface. Such emergent properties include: (i) an intrinsic angular momentum (beyond spin) that originates from the electron’s interaction with the crystalline lattice of ions, and (ii) the geometric Berry phase that an electron acquires in travelling around the Fermi surface. For some symmetry classes of metals, these properties are remarkably robust – unchanging under continuous deformations. Such robustness is a hallmark of a new generation of ‘topological metals’, whose recent discovery has revolutionized the field of condensed matter. I will describe how a well-known experimental technique (magnetic quantum oscillations) can be refined to unambiguously diagnose a topological metal from a conventional one.