  Cosmic-rays with energies up to $3\times10^{20}$ eV have been observed, as
have as have astrophysical neutrinos with energies above 1 PeV. In this talk, I
will discuss some of the unique phenomena that occur when particles with TeV
energies and above interact with matter. The emphasis will be on lepton
interactions. The cross-sections for electron bremsstrahlung and photon pair
conversion are suppressed at high energies, by the Landau-Pomeranchuk-Migdal
(LPM) effect, lengthening electromagnetic showers. At still higher energies
(above $10^{20}$ eV), photonuclear and electronuclear interactions dominate,
and showers become predominantly hadronic. Muons interact much less strongly,
so can travel long distances through solids before losing energy. Tau leptons
behave similarly, although their short livetime limits how far they can travel.
The hadronic interaction cross-section is believed to continue to increase
slowly with rising energy; measurements of cosmic-ray air showers support this
prediction.
