  Neutron stars contain matter in one of the densest forms found in the
Universe. This feature, together with the unprecedented progress in
observational astrophysics, makes such stars superb astrophysical laboratories
for a broad range of exciting physical studies. This paper gives an overview of
the phases of dense matter predicted to make their appearance in the cores of
neutron stars. Particular emphasis is put on the role of strangeness. Net
strangeness is carried by hyperons, K-mesons, H-dibaryons, and strange quark
matter, and may leave its mark in the masses, radii, moment of inertia,
dragging of local inertial frames, cooling behavior, surface composition, and
the spin evolution of neutron stars. These observables play a key role for the
exploration of the phase diagram of dense nuclear matter at high baryon number
density but low temperature, which is not accessible to relativistic heavy ion
collision experiments.
