  We review the main advantages and limitations of the kinetic exospheric and
fluid models of the solar wind (SW). We discuss the hydrostatic model imagined
by Chapman, the first supersonic hydrodynamic models published by Parker and
the first generation subsonic kinetic model proposed by Chamberlain. It is
shown that a correct estimation of the electric field as in the second
generation kinetic exospheric models developed by Lemaire and Scherer, provides
a supersonic expansion of the corona, reconciling the hydrodynamic and the
kinetic approach. The third generation kinetic exospheric models considers
kappa velocity distribution function (VDF) instead of a Maxwellian at the
exobase and in addition they treat a non-monotonic variation of the electric
potential with the radial distance; the fourth generation exospheric models
include Coulomb collisions based on the Fokker--Planck collision term.
Multi-fluid models of the solar wind provide a coarse grained description and
reproduce with success the spatio-temporal variation of SW macroscopic
properties (density, bulk velocity). The main categories of multi-fluid SW
models are reviewed: the 5-moment, or Euler, models, originally proposed by
Parker to describe the supersonic SW expansion; the 8-moment and 16-moment
fluid models, the gyrotropic approach with improved collision terms as well as
the gyrotropic models based on observed VDFs. The outstanding problem of
collisions, including the long range Coulomb encounters, is also discussed,
both in the kinetic and multi-fluid context. Although for decades the two
approaches have been seen as opposed, in this paper we emphasize their
complementary. The review of the kinetic and fluid models of the solar wind
contributes also to a better evaluation of the open questions still existent in
SW modeling and suggests possible future developments.
