Key concepts in time-dependent density-functional theory

We give an overview of the underlying concepts of time-dependent density-fu nctional theory. The basic relations between densities, potentials and init ial states, for time-dependent many-body systems are discussed. We obtain s ome new results concerning the invertability of response functions. Some fu ndamental difficulties associated with the time-dependent action principle are discussed and we show how these difficulties can be resolved by means o f the Keldysh formalism.