  Rotating neutron stars that support long-lived, non-axisymmetric deformations
known as mountains have long been considered potential sources of gravitational
radiation. However, the amplitude from such a source is very weak and current
gravitational-wave interferometers have yet to witness such a signal. The lack
of detections has provided upper limits on the size of the involved
deformations, which are continually being constrained. With expected
improvements in detector sensitivities and analysis techniques, there is good
reason to anticipate an observation in the future. This review concerns the
current state of the theory of neutron-star mountains. These exotic objects
host the extreme regimes of modern physics, which are related to how they
sustain mountains. We summarise various mechanisms that may give rise to
asymmetries, including crustal strains built up during the evolutionary history
of the neutron star, the magnetic field distorting the star's shape and
accretion episodes gradually constructing a mountain. Moving beyond the simple
rotating model, we also discuss how precession affects the dynamics and
modifies the gravitational-wave signal. We describe the prospects for detection
and the challenges moving forward.
