  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 $\bar{\nu}_{e}e$ 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 \v{C}erenkov detectors and large-area ground arrays.
