Neutrinos are unique cosmic messengers. Present attempts are directed to extend the window of cosmic neutrino observation from low energies (Sun, supernovae) to much higher energies. The aim is to study the most violent processes in the Universe which accelerate charged particles to highest energies, far beyond the reach of laboratory experiments on Earth. These processes must be accompanied by the emission of neutrinos. Neutrinos are electrically neutral and interact only weakly with ordinary matter; they thus propagate through the Universe without absorption or deflection, pointing back to their origin. Their feeble interaction, however, makes them extremely difficult to detect. The years 2008-2010 have witnessed remarkable steps in developing high energy neutrino telescopes. In 2010, the cubic-kilometre neutrino telescope IceCube at the South Pole has been completed. In the Mediterranean Sea the first-generation neutrino telescope ANTARES takes data since 2008, and efforts are directed towards KM3NeT, a telescope on the scale of several cubic kilometres. The next years will be key years for opening the neutrino window to the high energy Universe. With an instrumented volume of a cubic kilometre, IceCube is entering a region with realistic discovery potential. Discoveries or non-discoveries of IceCube will have a strong impact on the future of the field and possibly mark a "moment of truth". In this review, we discuss the scientific case for neutrino telescopes, describe the detection principle and its implementation in first- and second-generation installations and finally collect the existing physics results and the expectations for future detectors. We conclude with an outlook to alternative detection methods, in particular for neutrinos of extremely high energies.

84 pages, 49 figures. Accepted for publication in Progress in Particle and Nuclear Physics (PPNP)