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 × 1013 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 paper 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 behaviour 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.