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Neutron stars are fascinating astrophysical objects immersed in stronggravitational and electromagnetic fields, at the edge of our current theories.These stars manifest themselves mostly as pulsars, emitting a timely verystable and regular electromagnetic signal. Even though discovered almost fiftyyears ago, they still remain mysterious compact stellar objects. In thisreview, we summarize the most fundamental theoretical aspects of neutron starmagnetospheres and winds. The main competing models susceptible to explaintheir radiative properties like multi-wavelength pulse shapes and spectra andthe underlying physical processes such as pair creation and radiationmechanisms are scrutinized. A global but still rather qualitative pictureemerges slowly thanks to recent advances in numerical simulations on thelargest scales. However considerations about pulsar magnetospheres remainspeculative. For instance the exact composition of the magnetospheric plasma isnot yet known. Is it solely filled with a mixture of $e^\pm$~leptons, or doesit contain a non negligible fraction of protons and/or ions? Actually, is italmost entirely filled or mostly empty except for some small anecdotal plasmafilled regions? Answers to these questions will strongly direct the descriptionof the magnetosphere to seemingly contradictory results leading sometimes toinconsistencies. Nevertheless, account are given to the latest developments inthe theory of pulsar magnetospheres and winds, the existence of a possibleelectrosphere and physical insight obtained from related observationalsignatures of multi-wavelength pulsed emission.

Theory of pulsar magnetosphere and wind