A linear theory for traveling-wave tubes, based upon a normal mode expansion of a periodic slowwave circuit, was developed. The distinctive feature of the theory is that as many space harmonics as desired can be included in an analysis. If a full set of space harmonics are included, an infinite number of possible modesARE PREDICTED RATHER THAN THE FOUR MODES OF CONVENTIONAL THEORY. The theory reduces to the forward-wave amplifier theory of Pierce if only the dominant pair of space harmonics for forwardwave interaction are included, whereas it reduces to Johnson's theory of the backward-wave amplifier and backward-wave oscillator if only the dominant pair of space harmonics for backward-wave interaction are included. The normalmode expansion was also used to develop a nonlinear theory for traveling-wave tubes that allows both space harmonics and time harmonics to be taken into account. An analysis of the backward-wave oscillator was carried out using the nonlinear theory, assuming that only interaction with the backward-wave space harmonic at the fundamental frequency is significant. (Author).