The structure of the vacuum has become of general importance in modern physics. In this context the vacuum of quantum electrodynamics plays a special role because it changes drastically in strong external electric fields and these changes can be followed in a controllable way in the laboratory. It is reviewed how in supercritical electric fields the electron-positron-vacuum, i.e., the ground state of QED is changing from a neutral to a charged vacuum. The theoretical concepts, developed during the last 15 years, can be tested in collisions of very heavy ions, like U+U, U+Cm, etc. The important theoretical and experimental stages are described; in particular it is shown that the vacuum decay might have recently been experimentally verified: Sharp line structures in the positron spectra with pronounced threshold effects are the signals of this fundamental phase transition. Furthermore, the observed γ-ray and δ-electron spectra, the non-appearance of a massive line structure in undercritical systems like U+Ta and the quantitative understanding of the innershell ionization phenomena in giant quasimolecules (their agreement with theoretical predictions) give confidence into this interpretation. At the same time the positron line structures possibly indicate the existence of rather long living giant nuclear systems with Z ∼ 180–190. It is shown that most likely giant nuclear molecules have been formed as doorway states, but that other new not yet completely understood nuclear structure of giant complexes is indicated. The structure of the giant complexes can eventually be investigated spectroocopically by the positrons from the vacuum: The positron spectrum should not only show the vacuum decay line, but also Raman and Stokes satellites revealing the nuclear collective modes of the giant system formed. Various minima in the collective potential of giant systems should reveal the vacuum decay and their own local nuclear structure through corresponding classes of positron lines. The seemingly constancy with (Z T + Z P) of the principal positron line structures observed so far could indicate the formation of always the same giant object. Other possibilities to explain this effect (vacuum vibrations, excitation of the same nuclear mode in various giant systems and its supercritical conversion) are also discussed. In particular, the possibility of the creation and decay of a yet unknown particle causing constant positron line structure is shown to be in contradiction with atomic precision experiments. Finally the overcritical phenomenon in other areas of physics (e.g., overcritical gluo-electric fields, overcritical gravitational fields) is shortly discussed.