  We discuss in some detail the production of extremely high energy (EHE)
neutrinos with energies above 10^18 eV. The most certain process for producing
such neutrinos results from photopion production by EHE cosmic rays in the
cosmic background photon field. However, using assumptions for the EHE cosmic
ray source evolution which are consistent with results from the deep QSO survey
in the radio and X-ray range, the resultant flux of neutrinos from this process
is not strong enough for plausible detection. A measurable flux of EHE
neutrinos may be present, however, if the highest energy cosmic rays which have
recently been detected well beyond 10^20 eV are the result of the annihilation
of topological defects which formed in the early universe. Neutrinos resulting
from such decays reach energies of the grand unification (GUT) scale, and
collisions of superhigh energy neutrinos with the cosmic background neutrinos
initiate neutrino cascading which enhances the EHE neutrino flux at Earth. We
have calculated the neutrino flux including this cascading effect for either
massless or massive neutrinos and we find that these fluxes are conceivably
detectable by air fluorescence detectors now in development. The
neutrino-induced showers would be recognized by their starting deep in the
atmosphere. We evaluate the feasibility of detecting EHE neutrinos this way
using air fluorescence air shower detectors and derive the expected event rate.
Other processes for producing deeply penetrating air showers constitute a
negligible background.
