The modelling of pulsar magnetosphere and radiation

We review the recent advances in the pulsar high-energy $\gamma$-ray observation and the electrodynamics of the pulsar magnetospheres from the early vacuum model to the recent plasma-filled models by the numerical simulations. The numerical simulations have made the significant progresses toward the self-consistent modeling of the plasma-filled magnetosphere by including the particle acceleration and radiation. The current numerical simulations confirm a near force-free magnetosphere with the particle acceleration in the separatrix near the light cylinder and the current sheet outside the light cylinder, which can provide a good match to the recent high-energy $\gamma$-ray observations. The modeling of the combined multi-wavelength light curves, spectra, and polarization are expected to provide a stronger constrain on the geometry of the magnetic field lines, the location of the particle acceleration and the emission region, and the emission mechanism in the pulsar magnetospheres.