Although good progress has been made in the calculation of correlation energies from total energy expressions which are implicit functionals of the one-particle reduced density matrix, and explicit functionals of the natural orbitals (NOs) and their occupation numbers, a formulation of the calculation of excitation energies in this so-called density-matrix-functional theory (DMFT) is still lacking. In this paper we propose a time-dependent density-matrix- functional theory (TDDMFT). It is based on the equation of motion (EOM) for the 1-matrix P(s) (t) in the representation of the stationary NOs. In the final form of the EOM, the rate of change of the P(s) (t), P(s) (t) t, is determined by the commutator of the generalized time-dependent Fock matrix F(s) (t) with P(s) (t) plus an additional term D(s) (t). The matrix F(s) (t) determines the evolution of the NOs in the time-dependent one-electron Schrödinger equations, while D(s) (t) determines the time evolution of the NO occupations. With the neglect of the electron Coulomb correlation, the time-dependent one-electron equations for the NOs reduce to those for the Hartree-Fock (HF) orbitals of time-dependent HF (TDHF) theory. The coupled-perturbed equations of TDDMF response theory (TDDMFRT) are derived for the linear response of the 1-matrix δ P(s) (t) to a time-dependent perturbation δ vext (t) of the external potential. The frequency-dependent changes δ P (s), ij (ω) and δ P (s), kl (ω) are coupled through the coupling matrix Kijkl (ω), which is produced with the derivatives of F(s) (t) and D(s) (t) with respect to Pkl (t′). Based on the response equations, TDDMFRT eigenvalue equations are derived for the electron excitations ωq. © 2007 The American Physical Society.