Gravitational waves from isolated neutron stars

Neutron star interiors are a fantastic laboratory for high density physics in extreme environments. Probing this system with standard electromagnetic observations is, however, a challenging endeavour, as the radiation tends to be scattered by the outer layers and the interstellar medium. Gravitational waves, on the other hand, while challenging to detect, interact weakly with matter and are likely to carry a clean imprint of the high density interior of the star. In particular long lived, i.e. `continuous' signals from isolated neutron stars can carry a signature of deformations, possibly in crystalline exotic layers of the core, or allow to study modes of oscillation, thus performing gravitational wave asteroseismology of neutron star interiors. In this article we will review current theoretical models for continuous gravitational wave emission, and observational constraints, both electromagnetic and gravitational. Finally we will discuss future observational possibilities.