Magnetars are the most powerful compact objects in the stellar mass range observed in the Milky Way. Periodicity seen in magnetar persistent emission and also in the temporal tails of powerful giant flares detected from three soft gamma repeaters, and distinctive "spin-down" lengthening of the period have driven the paradigm that strongly-magnetized neutron stars constitute these fascinating sources. The steady X-ray emission includes both thermal atmospheric components and magnetospheric contributions that masquerade as hard X-ray "tails." This paper addresses observational and theoretical elements pertinent to the steady hard X-ray emission of magnetars, focusing on dissipative processes in their magnetospheres, and elements of Comptonization and polarization. It also discusses the action and possible signatures of exotic QED mechanisms like photon splitting and magnetic pair creation, and the quest for their observational vindication.