Neutron stars are currently the only known ‘laboratories’ that allow for
testing theories of the densest, cold matter in extreme conditions unattainable
at Earth. Description of the neutron star interiors - their equation of state
and, subsequently, their internal structure - is however largely unknown. Until
the birth of gravitational-wave astronomy, most of our knowledge came from
astronomical observations in electromagnetic waves, numerical simulations, but
also from terrestrial nuclear experiments, which provide an important link to the
low-density regime of nuclear physics. I will discuss how the electromagnetic
observations of neutron stars and gravitational-wave detections of inspiraling
binary neutron star systems can be used to constrain the equation of state
and possible exotic phases of matter. In addition to inspirals and mergers of
neutrons stars, there are currently a few proposed mechanisms that can trigger
radiation of long-lasting (continuous) gravitational radiation from neutron stars,
such as e.g., elastically and/or magnetically driven deformations: mountains
on the stellar surface supported by the elastic strain or magnetic field, free
precession, or unstable oscillation modes (e.g., the r-modes). I will review how
detection of such continuous gravitational radiation will allow us for the better
understanding of the neutron stars interiors, especially on their crust physics
and elastic properties at low temperatures.
