We present an analytical study of one-dimensional semiconductor superlattices in external electric fields, which may be time dependent. A number of general results for the (quasi)energies and eigenstates are derived. An equation of motion for the density matrix is obtained for a two-band model and the properties of the solutions are analyzed. An expression for the current is obtained. Finally, Zener tunneling in a two-band tight-binding model is considered. The present work gives the background and an extension of the theoretical framework underlying our recent Letter [J. Rotvig et al., Phys. Rev. Lett. 74, 1831 (1995)], where a set of numerical simulations was presented.