High-accuracy results for energy levels of heavy and superheavy atoms are presented. The projected (or no-virtual-pair) Dirac-Coulomb-Breit Hamiltonian serves as the starting point and defines the physical framework. One-electron four-component Dirac-Fock-Breit functions, similar in spirit to Hartree-Fock orbitals in the nonrelativistic formulation, are calculated first, followed by treatment of electron correlation. Correlation is included by the Fock-space coupled cluster method. The recent intermediate Hamiltonian approach makes it possible to use larger and more flexible P (model) spaces, thereby extending the range of applicability to states not accessible before. Accuracy is greatly improved for systems which can be treated by both methods, and the model space structure can be studied and pushed to convergence for the first time. Applications address mostly transition energies (ionization potentials, excitation energies, electron affinities) in various atoms. Very large basis sets, going up to l = 8, are used. High-l orbitals are particularly important for transitions involving f electrons. The outer 20-40 electrons of the atom are correlated. The Breit term is required for fine-structure splittings and for f transitions. Representative applications are described, including electron affinities of alkali atoms, obtained within 5 meV of known experimental values and providing the best estimate of the experimentally unknown EA of francium; the gold atom, with relativistic effects of 3-4 eV on transition energies; and Pr3+, where the many f2 levels are reproduced with great precision. The most exciting aspect of the high accuracy provided by the method is the ability to obtain reliable predictions for superheavy elements, where level ordering (and therefore chemistry) may differ from that of the lighter homologues. Thus, the ground state of eka-gold (E111) is 6d97s2, rather than the 6d107s expected from other group-11 elements; in Rf (E104), opposite effects of relativity and correlation lead finally to a 7s26d2 ground state, ∼0.3 eV below the 7s26d7p predicted by MCDF; eka-lead (E114), a potential member of the "island of stability" forecast by nuclear physics, is predicted to have ionization potentials higher than all other group-14 atoms except carbon; and eka-radon (E118) has a unique property for a rare gas, binding an electron with an affinity of -0.064(2) eV. QED corrections are calculated for the latter property, reducing the binding energy of the 8s electron by 0.0059(5) eV or 9%, the largest relative QED effect reported for a neutral or weakly ionized species.