  In this lecture, we give a first introduction to neutron stars, based on
fundamental physical principles. After outlining their outstanding macroscopic
properties, as obtained from observations, we infer the extreme conditions of
matter in their interiors. We then describe two crucial physical phenomena
which characterize compact stars, namely the gravitational stability of
strongly degenerate matter and the neutronization of nuclear matter with
increasing density, and explain how the formation and properties of neutron
stars are a direct consequence of the extreme compression of matter under
strong gravity. Finally, we describe how multi-wavelength observations of
different external macroscopic features (e.g. maximum mass, surface
temperature, pulsar glitches) can give invaluable information about the exotic
internal microscopic scenario: super-dense, isospin-asymmetric, superfluid,
bulk hadronic matter (probably deconfined in the most central regions) which
can be found nowhere else in the Universe. Indeed, neutrons stars represent a
unique probe to study the properties of the low-temperature, high-density
sector of the QCD phase diagram. Moreover, binary systems of compact stars
allow to make extremely precise measurements of the properties of curved
space-time in the strong field regime, as well as being efficient sources of
gravitational waves.
