Searches for continuous gravitational waves from neutron stars: A twenty-year retrospective

Seven years after the first direct detection of gravitational waves, from the collision of two black holes, the field of gravitational wave astronomy is firmly established. A first detection of continuous gravitational waves from rapidly-spinning neutron stars could be the field's next big discovery. I review the last twenty years of efforts to detect continuous gravitational waves using the LIGO and Virgo gravitational wave detectors. I summarise the model of a continuous gravitational wave signal, the challenges to finding such signals in noisy data, and the data analysis algorithms that have been developed to address those challenges. I present a quantitative analysis of 297 continuous wave searches from 80 papers, published from 2003 to 2022, and compare their sensitivities and coverage of the signal model parameter space.