We report new calculations of the cross sections for deeply inelastic neutrino-nucleon scattering at neutrino energies between $10^{9}\ev$ and $10^{21}\ev$. We compare with results in the literature and assess the reliability of our predictions. For completeness, we briefly review the cross sections for neutrino interactions with atomic electrons, emphasizing the role of the $W$-boson resonance in $\barν_{e}e$ interactions for neutrino energies in the neighborhood of $6.3\pev$. Adopting model predictions for extraterrestrial neutrino fluxes from active galactic nuclei, gamma-ray bursters, and the collapse of topological defects, we estimate event rates in large-volume water Čerenkov detectors and large-area ground arrays.

32 pages, 11 figures, uses RevTeX and boxedeps