(5 votes from 3 institutions)
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.