  Gravitational waves astronomy allows us to study objects and events invisible
in electromagnetic waves. It is crucial to validate the theories and models of
the most mysterious and extreme matter in the Universe: the neutron stars. In
addition to inspirals and mergers of neutrons stars, there are currently a few
proposed mechanisms that can trigger radiation of long-lasting gravitational
radiation from neutron stars, such as e.g., elastically and/or magnetically
driven deformations: mountains on the stellar surface supported by the elastic
strain or magnetic field, free precession, or unstable oscillation modes (e.g.,
the r-modes). The astrophysical motivation for continuous gravitational waves
searches, current LIGO and Virgo strategies of data analysis and prospects are
reviewed in this work.
