Cosmic ray showers and particle physics at energies ${10}^{15}$-${10}^{18}$ eV

Observations of extensive air showers generated by interactions of cosmic rays high in the atmosphere provide our only source of information about hadronic interactions above 1000 TeV. We review the current status of such experiments, discuss their implications for particle physics, and also note the astrophysical implications of the results. We place considerable emphasis on a description of the experiments and of the calculations required for their interpretation. We are motivated particularly by the relevance of existing air shower data both to the proposed new generation of accelerators, which will explore the region up to ∼1000 TeV, and to a new generation of air shower experiments, which has the potential to observe longitudinal development of individual showers with energies up to ${10}^{21}$ eV.