Fifteen years have past since the first beam collisions at the Large Hadron Collider (LHC) in CERN.
Not only the discovery of the Higgs particle, the LHC made a significant contribution to the cosmic-ray community.
Its designed collision energy $\sqrt{s}$=14TeV corresponds to the collision of a 10$^{17}$eV proton on a proton
at the rest frame, which is the energy range handled by the air shower observations.
The particle productions in the minimum-bias events and very-forward events have been extensively measured by the
various dedicated detectors at the LHC, and they serve crucial tests for the hadronic interaction models used in the
cosmic-ray air shower simulations.
In addition, collisions realized at various $\sqrt{s}$ are used to test the energy evolution of the hadronic interaction.
In this paper, we will review the key measurements at the LHC relevant to the air shower simulations especially focusing
on the forward measurements.
We will start the review from a quick outlook of some important concepts used in the high-energy and collider physics.