The project aims at the improved theoretical modelling and consistent interpretation of experimental data on very heavy and superheavy atomic nuclei with charge numbers Z greater than 82 and neutron numbers N beyond 126. These finite self-bound systems, many of which owe their very existence to quantal shell effects, exhibit a rich phenomenology of excitation and decay modes that are governed by the competition between the strong nuclear interaction, Coulomb repulsion, surface effects, and quantal shell structure of single-particle states. The available experimental data begin to reveal a consistent picture of their structure in terms of deformed shapes and shells, which at present, however, is not yet satisfactorily described by purely microscopic models. The main deficiency that has been clearly identified, and which is common to all presently available types of effective interactions, concerns the distance between single-particle levels near the Fermi energy. While global trends of observables are unaffected, the description of individual features of specific nuclei is in many cases lacking. The goal of our project is to arrive at an unprecedented level of accuracy for the theoretical description of very heavy and superheavy nuclei through the adjustment of an effective interaction containing qualitatively new and hitherto unused higher-order terms. The fit of its parameters will take into account information about relevant properties of states of heavy nuclei and be accompanied by an analysis of statistical errors. The resulting interactions will subsequently be employed in systematic symmetry-unrestricted self-consistent mean-field calculations of a wide spectrum of observables of interest addressed in in-beam gamma-ray and conversion-electron spectroscopy, implanted-ion decay spectroscopy, and laser spectroscopy. The results then can be used for the planning and evaluation of experiments at existing and future heavy-ion-beam facilities. We expect this project to make a decisive contribution to the progress in the theoretical description of the heaviest elements that will expand our understanding of these systems.