This is a review of neutrino astronomy anchored to the observational fact that Nature accelerates protons and photons to energies in excess of 10(20) and 10(13) eV, respectively. Although the discovery of cosmic rays dates back close to a century, we do not know how and where they are accelerated. There is evidence that the highest-energy cosmic rays are extra-galactic-they cannot be contained by our galaxy's magnetic field anyway because their gyroradius far exceeds its dimension. Elementary elementary-particle physics dictates a universal upper limit on their energy of 5 x 10(19) eV, the so-called Greisen-Kuzmin-Zatsepin cutoff; however, particles in excess of this energy have been observed by all experiments, adding one more puzzle to the cosmic ray mystery. Mystery is fertile ground for progress: we will review the facts as well as the speculations about the sources. There is a realistic hope that the oldest problem in astronomy will be resolved soon by ambitious experimentation: air shower arrays of 10(4) km(2) area, arrays of air Cerenkov detectors and, the subject of this review, kilometre-scale neutrino observatories. We will review why cosmic accelerators are also expected to be cosmic beam dumps producing associated high-energy photon and neutrino beams. We will work in detail through an example of a cosmic beam dump, gamma-ray bursts (GRBs). These are expected to produce neutrinos from MeV to EeV energy by a variety of mechanisms. We will also discuss active galaxies and GUT-scale remnants, two other classes of sources speculated to be associated with the highest-energy cosmic rays. GRBs and active galaxies are also the sources of the highest-energy gamma-rays, with emission observed up to 20 TeV, possibly higher. The important conclusion is that, independently of the specific blueprint of the source, it takes a kilometre-scale neutrino observatory to detect the neutrino beam associated with the highest-energy cosmic rays and gamma-rays. We also briefly review the ongoing efforts to commission such instrumentation.
