Pulsars (and their magnetospheres) have emerged in recent years as incredibly powerful probes of new fundamental physics, shedding light on everything from the properties of dark matter to the dynamics of early Universe. In this talk I will discuss a relatively new idea centered on how the electrodynamics of pulsar magnetospheres can be used to look for axions. In particular, I will argue that high-density clouds of gravitationally bound axions are generically expected to appear around pulsars for a broad range of axion parameter space. The densities in these axion clouds can be sufficiently large so as to overcome the inherently feeble nature of their interactions, giving rise to a variety of distinctive signatures, including: the production of sharp features in the radio band, and a short-lived periodic suppression of the pulsar's intrinsic radio emission. This is a young, but promising, field which offers immense complementarity with traditional laboratory searches for axions.