Although discovered almost forty years ago, pulsars still rank amongst the most fascinating astrophysical objects in the universe. They are believed to be strongly magnetised rotating neutron stars, whose existence had already been predicted soon after the discovery of the neutron in the beginning of the 30 s. However, relatively little progress has been made in understanding the fundamental physical mechanisms at work. Due to the lack of a satisfactory equation of state at very high densities, the internal structure of the neutron star remains enigmatic. Moreover, a global self-consistent picture of the close surrounding of the star, responsible for the emission of electromagnetic waves as well as for the launch of the pulsar wind and the energy loss due to interaction with the ambient medium, has not yet been proposed. Here, we give a brief review on the theory of pulsar's electrosphere (and magnetosphere) as well as on some recent developments. After a historical introduction, we recall some basic properties of these compact objects and the main observational characteristics, describing our present knowledge about pulsars. We then present our understanding of the pulsar magnetosphere, explaining in some detail a few theoretical ideas such as the polar cap and outer gap models. Some promising recent developments will be discussed: the electrospheric structure and the striped wind model.