  The properties of cosmic rays with energies above 10**6 GeV have to be
deduced from the spacetime structure and particle content of the air showers
which they initiate. In this review we summarize the phenomenology of these
giant air showers. We describe the hadronic interaction models used to
extrapolate results from collider data to ultra high energies, and discuss the
prospects for insights into forward physics at the LHC. We also describe the
main electromagnetic processes that govern the longitudinal shower evolution,
as well as the lateral spread of particles. Armed with these two principal
shower ingredients and motivation from the underlying physics, we provide an
overview of some of the different methods proposed to distinguish primary
species. The properties of neutrino interactions and the potential of
forthcoming experiments to isolate deeply penetrating showers from baryonic
cascades are also discussed. We finally venture into a terra incognita endowed
with TeV-scale gravity and explore anomalous neutrino-induced showers.
