Can QED be a UV-complete theory?

While still being the world’s most accurately tested physical theory, quantum electrodynamics (QED) appears to suffer from inconsistencies at highest energies. For instance, the extension of perturbation theory to high-energy scales is inhibited by the appearance of a Landau pole in the coupling as well as in correlation functions, indicating that the quest for a high-energy-complete version of QED is a nonperturbative problem. We discuss recent explorations of routes towards high-energy completeness, the non-perturbative fate of the Landau pole, the role of chiral symmetry and its possible dynamical or explicit breaking. Finally, we present first explorations of a new asymptotic-safety scenario in QED and provide evidence for the existence of various universality classes in the QED theory space.