Astrophysics with continuous gravitational waves

Direct detection of gravitational waves has become a powerful new tool of multi-messenger astrophysics. Apart from short-duration (transient) events, such as the inspirals and mergers of stellar-mass compact objects, we envisage other signal types of much longer duration-continuous gravitational waves. Traditionally associated with neutron star astrophysics, and hence with their largely unknown dense-matter interiors, continuous gravitational waves are now also entering other fields of astrophysics, namely, searches for dark matter, primordial black holes or exotic particles. Their long duration allows for qualitatively new possibilities, such as reproducible studies and tests of gravitational theory. This Review summarizes the results obtained in the recent O3 observing run of the LIGO-Virgo-KAGRA collaboration, the current status of the data analysis and the theoretical ideas related to astrophysical models. We show that O3 observations have started probing astrophysically relevant scenarios, and discuss how the improved sensitivity of the currently ongoing O4 observing run may offer a real possibility for a first detection of continuous gravitational waves. Continuous gravitational waves represent an exciting new frontier for multi-messenger astrophysics. This overview discusses their origins and the results obtained in the LIGO-Virgo-KAGRA O3 observing run, highlighting their potential for future studies.