  Ultra-high energy cosmic neutrinos are incisive probes of both astrophysical
sources and new TeV-scale physics. Such neutrinos would create extensive air
showers deep in the atmosphere. The absence of such showers implies upper
limits on incoming neutrino fluxes and cross sections. Combining the exposures
of AGASA, the largest existing ground array, with the exposure of the Fly's Eye
fluorescence detector integrated over all its operating epochs, we derive 95%
CL bounds that substantially improve existing limits. We begin with
model-independent bounds on astrophysical fluxes, assuming standard model cross
sections, and model-independent bounds on new physics cross sections, assuming
a conservative cosmogenic flux. We then derive model-dependent constraints on
new components of neutrino flux for several assumed power spectra, and we
update bounds on the fundamental Planck scale M_D in extra dimension scenarios
from black hole production. For large numbers of extra dimensions, we find M_D
> 2.0 (1.1) TeV for \mbhmin = M_D (5 M_D), comparable to or exceeding the most
stringent constraints to date.
