The structure of atoms, molecules and solids is mainly determined by static Coulomb forces, while the coupling to the quantized degrees of freedom of the electromagnetic field plays only a secondary role. Recently, it has been speculated that this general rule can be overcome in the context of cavity quantum electrodynamics (QED), where the coupling of dipoles to a single field mode can exceed the bare energy of the photon itself. Under these conditions, light-matter interactions become non-perturbative, as characterized by an effective fine-structure constant of order unity. In this seminar I will give a basic introduction to this extreme coupling regime of cavity QED and explain how vacuum-induced many-body effects can lead to novel ground state phases in QED, which are the opposite of what has been assumed so far. Beyond a purely fundamental interest, these general mechanisms can be important for potential applications, ranging from cavity-assisted chemistry to quantum technologies based on ultrastrongly coupled circuit QED systems.