Astroparticle Physics with High Energy Neutrinos: from AMANDA to IceCube

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: Statu