Black holes play a central role in astrophysics and in physics more generally. Candidate black holes are nearly ubiquitous in nature. They are found in the cores of nearly all galaxies, and appear to have resided there since the earliest cosmic times. They are also found throughout the galactic disk as companions to massive stars. Though these objects are almost certainly black holes, their properties are not very well constrained. We know their masses (often with errors that are factors of a few), and we know that they are dense. In only a handful of cases do we have information about their spins. Gravitational-wave measurements will enable us to rectify this situation. Focusing largely on measurements with the planned space-based detector LISA, I will describe how gravitational-wave measurements will allow us to measure both the masses and spins of black holes with percent-level accuracy even to high redshift, allowing us to track their growth and evolution over cosmic time. I will also describe how a special class of sources will allow us to measure the multipolar structure of candidate black hole spacetimes. This will make it possible to test the no-hair theorem, and thereby to test the hypothesis that black hole candidates are in fact black holes are described by general relativity.