  Cosmic-rays with energies up to $3\times10^{20}$ eV have been observed. The
nuclear composition of these cosmic rays is unknown but if the incident nuclei
are protons then the corresponding center of mass energy is $\sqrt{s_{nn}} =
700$ TeV. High energy muons can be used to probe the composition of these
incident nuclei. The energy spectra of high-energy ($>$ 1 TeV) cosmic ray
induced muons have been measured with deep underground or under-ice detectors.
These muons come from pion and kaon decays and from charm production in the
atmosphere.
  Terrestrial experiments are most sensitive to far-forward muons so the
production rates are sensitive to high-$x$ partons in the incident nucleus and
low-$x$ partons in the nitrogen/oxygen targets. Muon measurements can
complement the central-particle data collected at colliders. This paper will
review muon production data and discuss some non-perturbative (soft) models
that have been used to interpret the data. I will show measurements of TeV muon
transverse momentum ($p_T$) spectra in cosmic-ray air showers from MACRO, and
describe how the IceCube neutrino observatory and the proposed Km3Net detector
will extend these measurements to a higher $p_T$ region where perturbative QCD
should apply. With a 1 km$^2$ surface area, the full IceCube detector should
observe hundreds of muons/year with $p_T$ in the pQCD regime.
