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 analysis one has to face the conclusion that there is currently no valid foundation for TDDFT, and expectations of finding a remedy here are hardly justified.