High-energy (> 100 MeV) neutrino astrophysics enters an era of opportunity and discovery as the sensitivity of detectors approaches astrophysically relevant flux levels. We review the major challenges for this emerging field, among which the nature of dark matter, the origin of cosmic rays, and the physics of extreme objects such as active galactic nuclei, gamma-ray bursts, pulsars, and supernova remnants are of prime importance. Variable sources at cosmological distances allow the probing of neutrino propagation properties over baselines up to about 20 orders of magnitude larger than those probed by terrestrial long-baseline experiments. We review the possible astrophysical sources of high-energy neutrinos, which also act as an irreducible background to searches for phenomena at the electroweak and grand-unified-theory symmetry-breaking scales related to possible supersymmetric dark matter and topological defects. Neutrino astronomy also has the potential to discover previously unimagined high-energy sources invisible in other channels and provides the only means for direct observations of the early universe prior to the era of decoupling of photons and matter. We conclude with a discussion of experimental approaches and a short report on present projects and prospects. We look forward to the day when it will be possible to see the universe through a new window in the light of what may be its most numerous particle, the elusive neutrino.