  Cosmic rays with energies beyond the Greisen-Zatsepin-Kuzmin `cutoff' at
$\sim 4 \times 10^{10}$ GeV pose a conundrum, the solution of which requires
either drastic revision of our astrophysical understanding, or new physics
beyond the Standard Model. Nucleons of such energies must originate within the
local supercluster in order to avoid excessive energy losses through photopion
production on the cosmic microwave background. However they do not point back
towards possible nearby sources, e.g. the active galaxy Cen A or M87 in the
Virgo cluster, so such an astrophysical origin requires intergalactic magnetic
fields to be a hundred times stronger than previously believed, in order to
isotropise their arrival directions. Alternatively the primaries may be high
energy neutrinos, say from distant gamma-ray bursts, which annihilate on the
local relic background neutrinos to create ``Z-bursts''. A related possibility
is that the primary neutinos may initiate the observed air showers directly if
their interaction cross-sections are boosted to hadronic strength through
non-perturbative physics such as TeV-scale quantum gravity. Or the primaries
may instead be new strongly interacting neutral particles with a longer mean
free path than nucleons, coming perhaps from distant BL-Lac objects or FR-II
radio galaxies. Yet another possibility is that Lorentz invariance is violated
at high energies thus suppressing the energy loss processes altogether. The
idea that has perhaps been studied in most detail is that such cosmic rays
originate from the decays of massive relic particles (``wimpzillas'') clustered
as dark matter in the galactic halo. All these hypotheses will soon be
critically tested by the Pierre Auger Observatory, presently under construction
in Argentina, and by proposed satellite experiments such as EUSO.
