  Formed in the aftermath of gravitational core-collapse supernova explosions,
neutron stars are unique cosmic laboratories for probing the properties of
matter under extreme conditions that cannot be reproduced in terrestrial
laboratories. The interior of a neutron star, endowed with the highest magnetic
fields known and with densities spanning about ten orders of magnitude from the
surface to the centre, is predicted to exhibit various phases of dense strongly
interacting matter, whose physics is reviewed in this chapter. The outer layers
of a neutron star consist of a solid nuclear crust, permeated by a neutron
ocean in its densest region, possibly on top of a nuclear "pasta" mantle. The
properties of these layers and of the homogeneous isospin asymmetric nuclear
matter beneath constituting the outer core may still be constrained by
terrestrial experiments. The inner core of highly degenerate, strongly
interacting matter poses a few puzzles and questions which are reviewed here
together with perspectives for their resolution. Consequences of the
dense-matter phases for observables such the neutron-star mass-radius
relationship and the prospects to uncover their structure with modern
observational programmes are touched upon.
