Time-dependent density-functional theory (TDDFT) is deemed to be a formally rigorous way of dealing with the time-evolution of a many-electron system at the level of electron densities rather than the underlying wavefunctions, which in turn provides an, in principle, exact density-based approach to the treatment of electron excitations in atoms and molecules. This claim has not remained unchallenged, and a detailed account of the relevant criticism is given in this paper. In view of our …