  Neutron stars are fascinating astrophysical objects immersed in strong
gravitational and electromagnetic fields, at the edge of our current theories.
These stars manifest themselves mostly as pulsars, emitting a timely very
stable and regular electromagnetic signal. Even though discovered almost fifty
years ago, they still remain mysterious compact stellar objects. In this
review, we summarize the most fundamental theoretical aspects of neutron star
magnetospheres and winds. The main competing models susceptible to explain
their radiative properties like multi-wavelength pulse shapes and spectra and
the underlying physical processes such as pair creation and radiation
mechanisms are scrutinized. A global but still rather qualitative picture
emerges slowly thanks to recent advances in numerical simulations on the
largest scales. However considerations about pulsar magnetospheres remain
speculative. For instance the exact composition of the magnetospheric plasma is
not yet known. Is it solely filled with a mixture of $e^\pm$~leptons, or does
it contain a non negligible fraction of protons and/or ions? Actually, is it
almost entirely filled or mostly empty except for some small anecdotal plasma
filled regions? Answers to these questions will strongly direct the description
of the magnetosphere to seemingly contradictory results leading sometimes to
inconsistencies. Nevertheless, account are given to the latest developments in
the theory of pulsar magnetospheres and winds, the existence of a possible
electrosphere and physical insight obtained from related observational
signatures of multi-wavelength pulsed emission.
