Wide bandwidth vacuum tubes employ the interaction between a travelling electromagnetic wave and a stream of electrons. The electromagnetic structures must support electromagnetic waves whose phase velocity is less than the velocity of light. The essential properties of uniform slow-wave structures are derived from Maxwell’s equations. Virtually all practical slow-wave structures are periodic in space and their properties can be investigated using equivalent circuits. The r.f. electric field of the structure which interacts with the electrons can be expanded as a set of space harmonics using Fourier analysis. The properties of planar slow-wave structures in the form of meander and ladder lines can be explored using equivalent circuits whose parameters have been calculated from static field solutions. The properties of helix slow-wave structures can be understood using solutions of Maxwell’s equations but it is difficult to get accurate numerical results in this way. A simple alternative, which yields useful results, is to use equivalent circuit analysis similar to that for planar structures. This method can also be applied to the ring-bar structure. High power structures based on folded waveguides and coupled cavities can be modelled using equivalent circuits. Methods of measuring the properties of slow-wave structures are discussed.