We discuss recent progress in hadronic interaction models, applied to cosmic ray air shower simulations and to heavy ion and proton-proton scattering at RHIC and LHC. Cosmic rays at the highest possible energies are so rare that they can only be observed via the billions of secondary particles in the so-called air showers — cascades of particles, multiplying themselves due to successive interactions in the atmosphere. There is first of all the hadronic cascade, composed of hadrons, produced in hadron + air collisions. These hadrons are mainly pions, but also kaons, and protons, where the latter ones will play an important role in the later discussions. The charged pions decay preferentially into muons and neutrinos: π − → μ − ¯ νµ and π + → μ + νµ ,w ith a lifetime of 2.6 × 10 −8 s. The neutral pions, however, decay almost exclusively into two photons, with a lifetime of 0.8 × 10 −16 s — in other words immediately, before having a chance to scatter. The two photons of the neutral pion decay initiate a so-called electromagnetic cascade of photons and electrons, with the main processes at high energies being pair creation and Bremsstrahlung. The main observables are electrons (from the electromagnetic cascade), muons (from the decays of hadrons which do NOT decay into photons), and also fluorescence lights, radio signals, and so on. We will focus here on the electrons and the muons.