This thesis is concerned with the mathematical representation of, and the analysis of travelling waves on, power system transmission lines. The transmission lines are described mathematically by the Telegrapher's equations. Two variants are considered in detail; where the equation coefficients are constant and where they are functions of frequency. A third variant where the coefficients are functions of the line voltage is introduced but not given a detailed treatment. The travelling wave analysis consists of finding solutions of the Telegrapher's equations, bearing in mind the characteristics of the transmission lines, the equipment to which they will be connected and the nature of the signals likely to be experienced in power system transient situations. The constant coefficient Telegrapher's equations form the basis of the 'Linear Transmission Line Analysis', the first part of this thesis. The assumptions involved in this representation and the approximations that they introduce are considered. The many techniques available for solution are critically reviewed. Even for this, the simplest travelling wave representation of a power system transmission line, solution of transient problems can introduce considerable difficulties. Application of the method of characteristics to the solution of transient propagation problems is studied in detail. This method of solution is shown to have a much wider application than has generally been recognised. Its application has led to a number of new solutions which are shown to be superior to those in current use. Both single circuit and mutually coupled multiple circuit transmission lines have been considered. In order to facilitate possible future extension of this work into the area of nonlinear transmission special consideration has been given to the direct application of time domain solution methods to transient propagation problems. The Telegrapher's equations with frequency dependent coefficients form the second part of this thesis. Literature in this subject area is both copious and scattered making it a time-consuming process to attain a current knowledge of this work. To alleviate this situation a critical summary and discussion of significant results in both the formulation of these effects and solution of the resulting equations is given. Presentation is detailed where necessary. In a number of instances the work being discussed is developed beyond that originally given. A brief concluding section considers the current state of development towards a comprehensive first order power system transmission line representation.