  The properties of hyperheavy nuclei and the extension of nuclear landscape to
hyperheavy nuclei are extensively studied within covariant density functional
theory. Axial reflection symmetric and reflection asymmetric relativistic
Hartree-Bogoliubov (RHB) calculations are carried out. The role of triaxiality
is studied within triaxial RHB and triaxial relativistic mean field + BCS
frameworks. With increasing proton number beyond Z~130 the transition from
ellipsoidal-like nuclear shapes to toroidal ones takes place. The description
of latter shapes requires the basis which is typically significantly larger
than the one employed for the description of ellipsoidal-like shapes. Many
hyperheavy nuclei with toroidal shapes are expected to be unstable towards
multifragmentation. However, three islands of stability of spherical hyperheavy
nuclei have been predicted for the first time in Ref. [1]. Proton and neutron
densities, charge radii, neutron skins and underlying shell structure of the
nuclei located in the centers of these islands have been investigated in
detail. Large neutron shell gaps at N=228, 308 and 406 define approximate
centers of these islands in neutron number. On the contrary, large proton gap
appear only at Z=154 in the (Z~156, N~310) island. As a result, this is the
largest island of stability of spherical hyperheavy nuclei found in the
calculations. The calculations indicate the stability of the nuclei in these
islands with respect of octupole and triaxial distortions. Fission barriers in
neutron-rich superheavy nuclei are studied in triaxial RHB framework; the
impact of triaxiality on the heights of fission barriers is substantial in some
parts of this region. Based on the results obtained in the present work, the
extension of nuclear landscape to hyperheavy nuclei is provided.
