  Instability of electron-positron vacuum in strong electric fields is studied.
First, falling to the Coulomb center is discussed at $Z>137/2$ for a spinless
boson and at $Z>137$ for electron. Then, focus is concentrated on description
of deep electron levels and spontaneous positron production in the field of a
finite-size nucleus with the charge $Z>Z_{\rm cr}\simeq 170$. Next, these
effects are studied in application to the low-energy heavy-ion collisions.
Then, we consider phenomenon of "electron condensation" on levels of upper
continuum crossed the boundary of the lower continuum $\epsilon =-m$ in the
field of a supercharged nucleus with $Z\gg Z_{\rm cr}$. Finally, attention is
focused on many-particle problems of polarization of the QED vacuum and
electron condensation at ultra-short distances from a source of charge. We
argue for a principal difference of cases, when the size of the source is
larger than the pole size $r_{\rm pole}$, at which the dielectric permittivity
of the vacuum reaches zero, and smaller $r_{\rm pole}$. Some arguments are
presented in favor of the logical consistency of QED. All problems are
considered within the same relativistic semiclassical approach.
