Covariant density functional theory (CDFT) is a modern theoretical tool for the description of nuclear structure phenomena. Different physical observables of the ground and excited states in even-even nuclei have been studied within the CDFT framework employing three major classes of the state-of-the-art covariant energy density functionals. The global assessment of the accuracy of the description of the ground state properties and systematic theoretical uncertainties of atomic nuclei have been investigated. Large-scale axial relativistic Hartree-Bogoliubov (RHB) calculations are performed for all Z < 106 even-even nuclei between the two-proton and two-neutron drip lines. The sources of theoretical uncertainties in the prediction of the two-neutron drip line are analyzed in the framework of CDFT. We concentrate on single-particle and pairing properties as potential sources of these uncertainties. The major source of these uncertainties can be traced back to the differences in the underlying single-particle structure of the various CEDFs. A systematic search for axial octupole deformation in the actinides and superheavy nuclei with proton numbers Z = 88 - 126 and neutron numbers from two-proton drip line up to N = 210 has been performed in CDFT. The nuclei in the Z ~ 96, N ~ 196 region of octupole deformation have been investigated in detail and their systematic uncertainties have been quantified. The structure of superheavy nuclei has been reanalyzed with inclusion of quadrupole deformation. Theoretical uncertainties in the predictions of inner fission barrier heights in superheavy elements have been investigated in a systematic way. The correlations between global description of the ground state properties and nuclear matter properties have been studied. It was concluded that the strict enforcement of the constraints on the nuclear matter properties (NMP) defined in Ref. [1] will not necessary lead to the functionals with good description of ground state properties. The different aspects of the existence and stability of hyperheavy nuclei have been investigated. For the first time, we demonstrate the existence of three regions of spherical hyperheavy nuclei centered around (Z ~ 138, N ~ 230), (Z ~ 156, N ~ 310) and (Z ~ 174, N ~ 410) which are expected to be reasonably stable against spontaneous fission.