This is a review of neutrino astronomy anchored to the observational fact that Nature accelerates protons and photons to energies in excess of 10²⁰ and 10¹³ 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 5×10¹⁹ 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 10⁴ km² 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, γ-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 γ-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 γ-rays. We also briefly review the ongoing efforts to commission such instrumentation.