  Kilometer-scale neutrino detectors such as IceCube are discovery instruments
covering nuclear and particle physics, cosmology and astronomy. Examples of
their multidisciplinary missions include the search for the particle nature of
dark matter and for additional small dimensions of space. In the end, their
conceptual design is very much anchored to the observational fact that Nature
produces protons and photons with energies in excess of $10^{20}$\,eV and
$10^{13}$\,eV, respectively. The puzzle of where and how Nature accelerates the
highest energy cosmic particles is unresolved almost a century after their
discovery. The cosmic ray connection sets the scale of cosmic neutrino fluxes.
In this context, we discuss the first results of the completed AMANDA detector
and the science reach of its extension, IceCube. Similar experiments are under
construction in the Mediterranean. Neutrino astronomy is also expanding in new
directions with efforts to detect air showers, acoustic and radio signals
initiated by super-EeV neutrinos. The outline of this review is as follows:
  Introduction
  Why Kilometer-Scale Detectors?
  Cosmic Neutrinos Associated with the Highest Energy Cosmic Rays
  High Energy Neutrino Telescopes: Methodologies of Neutrino Detection
  High Energy Neutrino Telescopes: Status
