  With the doors beginning to swing open on the new gravitational wave
astronomy, this review provides an up-to-date survey of the most important
physical mechanisms that could lead to emission of potentially detectable
gravitational radiation from isolated and accreting neutron stars. In
particular we discuss the gravitational wave-driven instability and
asteroseismology formalism of the f- and r-modes, the different ways that a
neutron star could form and sustain a non-axisymmetric quadrupolar "mountain"
deformation, the excitation of oscillations during magnetar flares and the
possible gravitational wave signature of pulsar glitches. We focus on progress
made in the recent years in each topic, make a fresh assessment of the
gravitational wave detectability of each mechanism and, finally, highlight key
problems and desiderata for future work.
