We describe the status of the problem of the electron structure of superheavy atoms with nuclear charge Z > Z[sub]c[/sub] ; here Z[sub]c[/sub]≈170 is the critical value of the nuclear charge, at which the energy of the ground state of the 1S[sub]1/2 [/sub]electron reaches the limit of the lower continuum of the solutions of the Dirac equation (&z.ele; = - m[sub]e[/sub]c[sup]2[/sup]) . We discuss the dependence of Z[sub]c[/sub] on the nuclear radius R and on the character of the distribution of the electric charge inside the nucleus, and also the form of the wave functions at Z close to Z[sub]c[/sub] . Owing to the Coulomb barrier , the state of the electron remains localized at Z > Z[sub]c[/sub] , in spite of the fact that its energy approaches the continuum. An analysis of the polarization of the vacuum in a strong Coulomb field shows that a bare nucleus with supercritical charge Z > Z[sub]c[/sub] produces spontaneously two positrons and, in addition a charge density with a total of two units of negative charge in the vacuum. The distribution of this density is localized in a region of dimension r ≈ ħ/m[sub]e[/sub]c at the nucleus. The possibility of experimentally observing the effect of quasistatic production of positrons in the collision of two bare uranium nuclei (i.e., without electrons) is discussed. A brief review is presented of work on the motion of levels with increasing depth of the potential well in other relativistic equations (Kelin-Gordon, Proca, etc.).