Magnetars are the most luminous compact objects in the stellar mass range observed in the Milky Way, with giant flares of hard X-ray power ~10^45 erg/sec being detected from three soft gamma repeaters in the Galactic neighborhood. Periodicity seen in magnetar persistent emission, and a distinctive spin-down lengthening of this 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 are manifested 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 the exotic and fundamental QED mechanisms of photon splitting and magnetic pair creation, and the quest for their observational vindication.
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