Testing the no-hair theorem with observations of black holes in the electromagnetic spectrum

According to the general-relativistic no-hair theorem, astrophysical black holes depend only on their masses and spins and are uniquely described by the Kerr metric. Mass and spin are the first two multipole moments of the Kerr spacetime and completely determine all other moments. The no-hair theorem can be tested by measuring potential deviations from the Kerr metric which alter such higher-order moments. In this review, I discuss tests of the no-hair theorem with current and future observations of such black holes across the electromagnetic spectrum, focusing on near-infrared observations of the supermassive black hole at the Galactic center, pulsar-timing and very-long baseline interferometric observations, as well as x-ray observations of fluorescent iron lines, thermal continuum spectra, variability, and polarization.