  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.
