  The properties of toroidal hyperheavy even-even nuclei and the role of
toroidal shell structure are extensively studied within covariant density
functional theory. The general trends in the evolution of toroidal shapes in
the $Z\approx 130-180$ region of nuclear chart are established for the first
time. These nuclei are stable with respect of breathing deformations. The most
compact fat toroidal nuclei are located in the $Z\approx 136, N\approx 206$
region of nuclear chart, but thin toroidal nuclei become dominant with
increasing proton number and on moving towards proton and neutron drip lines.
The role of toroidal shell structure, its regularity, supershell structure,
shell gaps as well as the role of different groups of the pairs of the orbitals
in its formation are investigated in detail. The lowest in energy solutions at
axial symmetry are characterized either by large shell gaps or low density of
the single-particle states in the vicinity of the Fermi level in at least one
of the subsystems (proton or neutron). Related quantum shell effects are
expected to act against the instabilities in breathing and sausage deformations
for these subsystems. The investigation with large set of covariant energy
density functionals reveals that substantial proton $Z=154$ and 186 and neutron
$N=228$, 308 and 406 spherical shell gaps exist in all functionals. The nuclei
in the vicinity of the combination of these particle numbers form the islands
of stability of spherical hyperheavy nuclei. The study suggests that the
$N=210$ toroidal shell gap plays a substantial role in the stabilization of fat
toroidal nuclei.
