  In magnetic fields stronger than B_Q = 4.4 X 10^13 Gauss, an electron's
Landau excitation energy exceeds its rest energy. I review the physics of this
strange regime and some of its implications for the crusts and magneto- spheres
of neutron stars. In particular, I describe how ultra-strong fields
  >> render the vacuum birefringent and capable of distorting and magnifying
images ("magnetic lensing");
  >> change the self-energy of electrons: as B increases they are first
slightly lighter than $m_e$, then slightly heavier;
  >> cause photons to rapidly split and merge with each other;
  >> distort atoms into long, thin cylinders and molecules into strong,
polymer-like chains;
  >> enhance the pair density in thermal pair-photon gases; >> strongly
suppress photon-electron scattering, and >> drive the vacuum itself unstable,
at extremely large B.
  In a concluding section, I discuss recent observations of the spindown
histories of soft gamma repeaters and anomalous X-ray pulsars. The magnetar
model gives a promising framework for understanding these data.
