The time-dependent density-functional theory of Runge and Gross [Phys. Rev. Lett. 52, 997 (1984)] is reexamined on the basis of its limitations, and the criticisms raised by Xu and Rajagopal [Phys. Rev. A 31, 2682 (1985)] are addressed, within the imposition of natural boundary conditions of vanishing density and potential at infinity. Also, for a single-particle system characterized by an arbitrary time-dependent potential, the uniqueness of the density-to-potential mapping is established explicitly for both bound and scattering states.