Gravitational waves have now been detected from the coalescence of around 100 black hole and neutron star binary systems. In contrast, the search for long-lived “continuous” gravitational wave emission from a single spinning neutron star is ongoing, with a first detection hopefully coming soon. The detection of such signals offers the prospect of probing matter at extremes of density, temperature and magnetic field strength inaccessible to terrestrial experimenters. However, there are large uncertainties in the strength and frequency content of such emission. In this talk I discuss the difficulties in searching for these signals, and describe the most important theoretical issues in modelling them.