  We review electrodynamics of rotating magnetized neutron stars, from the
early vacuum model to recent numerical experiments with plasma-filled
magnetospheres. Significant progress became possible due to the development of
global particle-in-cell simulations which capture particle acceleration,
emission of high-energy photons, and electron-positron pair creation. The
numerical experiments show from first principles how and where electric gaps
form, and promise to explain the observed pulsar activity from radio waves to
gamma-rays.
