Aim. This article presents a review of the nonstationary (time-domain) discrete theory of excitation of periodic electromagnetic structures and discusses applications of the theory for simulation of traveling-wave tube (TWT) microwave power amplifiers with slow-wave structures (SWS) of different kind. Methods. The discrete theory is based on a representation of a periodic SWS as a chain of coupled cells. However, these cells are not identical to periods of the structure, and each cell is coupled with not only nearest neighbors, but, in general, with all the other cells. The discrete theory allows useful reformulation of Maxwell equations and simplifies simulation of electromagnetic wave propagation through a periodic structure by a great degree-of-freedom reduction. In this paper, we present the derivation of the basic equations of the discrete model from Maxwell equations and investigate the beam-wave interaction processes by numerical simulation. Results. Derivation of the discrete theory equations in its original form proposed by SP Kuznetsov is presented. The results of simulation of the C-band coupled-cavity (CC) TWT are considered, including complicated transients, which accompany spurious self-excitation near cut-off. Further developments of the discrete theory including the Hamiltonian formalism are discussed. The Hamiltonian discrete model is applied for simulation of the 170-W Ku-band helix TWT. The results of simulations are in good agreement with the experimental measurements. Conclusion. The discrete theory proposed by SP Kuznetsov in 1980 is a powerful tool for modeling of electromagnetic wave propagation in various periodic slow-wave structures. It allows development of computer codes for time-domain simulation of TWTs, which are promising tools that bears several advantages for industrial and research activities.