Title: Multimessenger observations and the science enabled: Continuous waves and their progenitors, equation of state of dense matter

Abstract: Rotating and oscillating neutron stars can give rise to long-lived Continuous Gravitational Waves (CGWs). Despite many years of searching, the detection of such a CGW signal remains elusive. In this article we describe the main astrophysical uncertainties regarding such emission, and their relation to the behaviour of matter at extremely high density. We describe the main challenges in searching for CGWs, and the prospects of detecting them using third-generation gravitational wave detectors. We end by describing some pressing issues in the field, whose resolution would help turn the detection and exploitation of CGWs into reality.