  The observation of cosmic--ray events above the Greisen-Kuzmin-Zatsepin (GZK)
cutoff of $\sim 5\times 10^{19}$ eV challenges orthodox modeling. We discuss a
possible solution which uses standard hot Big Bang cosmology and Standard Model
(SM) particle physics augmented only by $\lsim $ eV neutrino masses as
suggested by solar, atmospheric, and terrestrial neutrino detection. In this
scheme, cosmic ray neutrinos from distant, highest-energy sources annihilate
resonantly on the relic-neutrino background to produce Z-bosons. The
highly-boosted ($\gamma_Z\sim 10^{11}$) Z's instantly decay, producing
"Z-bursts" of highly-collimated jets of hadrons and photons. The burst content
includes, on average, twenty photons and two nucleons with super-GZK energies.
We show that the probability for each neutrino with energy within a fraction
$\Gamma_Z/M_Z$ of the resonant value $E_R=4(eV/m_\nu)\times 10^{21}$ eV to
annihilate within the halo of our galactic supercluster is likely within an
order of magnitude of 1%. Depending on the magnitude of the cosmic neutrino
flux above 10^{20} eV, this "local" rate for primary production may be high
enough to produce the cosmic ray events observed above the GZK cutoff. Several
tests of this Z-burst hypothesis for generating super-GZK events are presented,
including (i) a new cutoff energy at E_R; (ii) a large $\gamma/p$ ratio for
primaries near the upper end of the spectrum; (iii) directional pairing of
events, and pointing to their cosmic sources; and (iv) a neutrino flux above
the GZK cutoff energy which is possibly measurable directly in proposed $\gsim
10^{11}$ ton cosmic-ray detectors.
