Extremely high-energy particles traveling through matter (the atmosphere, quark-gluon plasmas, whatever) lose energy by showering -- a sequence of splittings via hard bremsstrahlung or pair production. At high enough energy, the "duration" of each individual splitting becomes longer than the mean free time for collisions with the medium, giving rise to an interference effect known as the Landau-Pomeranhcuk-Migdal (LPM) effect, which drastically changes the rate of shower development and so of energy loss. For QED, this effect was realized theoretically in the 1950s in the context of cosmic ray showers. For QCD, the formalism was first worked out in the 1990s, with applications to jet energy loss in quark-gluon plasmas. I will review the LPM effect, more recent developments in the QCD case, and why it is still an extremely rich and interesting field theory problem.