  We review our knowledge on the properties of the nuclear medium that have
been studied, along many years, on the basis of many-body theory, laboratory
experiments and astrophysical observations. First we consider the realm of
phenomenological laboratory data and astrophysical observations, and the hints
they can give on the characteristics that the nuclear medium should possess.
The analysis is based on phenomenological models, that however have a strong
basis on physical intuition and an impressive success. More microscopic models
are also considered, and it is shown that they are able to give invaluable
information on the nuclear medium, in particular on its Equation of State. The
interplay between laboratory experiments and astrophysical observations are
particularly stressed, and it is shown how their complementarity enriches
enormously our insights into the structure of the nuclear medium. We then
introduce the nucleon-nucleon interaction and the microscopic many-body theory
of nuclear matter, with a critical discussion about the different approaches
and their results. The Landau Fermi Liquid theory is introduced and briefly
discussed. As illustrative example, we discuss neutron matter at very low
density, and it is shown how it can be treated within the many-body theory. A
section is dedicated to the pairing problem. The connection with nuclear
structure is then discussed, on the basis of the Energy Density Functional
method. The possibility to link the physics of exotic nuclei and the
astrophysics of neutron stars is particularly stressed. Finally we discuss the
thermal properties of the nuclear medium, in particular the liquid-gas phase
transition and its connection with the phenomenology on heavy ion reactions and
the cooling evolution of neutron stars. The presentation has been taken for
non-specialists and possibly for non-nuclear physicists.
