  There has recently been growing evidence for the existence of neutron stars
possessing magnetic fields with strengths that exceed the quantum critical
field strength of $4.4 \times 10^{13}$ G, at which the cyclotron energy equals
the electron rest mass. Such evidence has been provided by new discoveries of
radio pulsars having very high spin-down rates and by observations of bursting
gamma-ray sources termed magnetars. This article will discuss the exotic
physics of this high-field regime, where a new array of processes becomes
possible and even dominant, and where familiar processes acquire unusual
properties. We review the physical processes that are important in neutron star
interiors and magnetospheres, including the behavior of free particles, atoms,
molecules, plasma and condensed matter in strong magnetic fields, photon
propagation in magnetized plasmas, free-particle radiative processes, the
physics of neutron star interiors, and field evolution and decay mechanisms.
Application of such processes in astrophysical source models, including
rotation-powered pulsars, soft gamma-ray repeaters, anomalous X-ray pulsars and
accreting X-ray pulsars will also be discussed. Throughout this review, we will
highlight the observational signatures of high magnetic field processes, as
well as the theoretical issues that remain to be understood.
