The aim of tbli tract is to give an account of the theory of topological vector spaces. There are many branches of functional analysis which depend upon this theory; it forms, for example, the background for the theory of distributions. In addition, it often clarinet results in the theory of normed spaces, especially those concerned with the weak topology, to regard them as particular cases of more general results about topological vector spaces. In this tract we have tried to present the main theorems as simply and directly as is possible without sacrificing any of their force or generality. It has been our purpose to make the basic ideas and facts of the subject easily accessible. With this in mind, we have assumed only a minimum of knowledge of general topology and linear algebra to start with; the details of what is assumed are to be found at the beginning of the first chapter. After this, new topological concepts am introduced when they are needed. The first four or five chapters contain the fundamental results; here the pace is leisurely. Some of these could have been deduced more concisely from general topological theorems, but we have preferred to keep the treatment self-contained and establish them directly for topological vector spaces, or even for locally convex spaces. From the sixth chapter onwards the proofs are more condensed; here we have allowed ourselves to be guided to some extent by our own interests in the choice of subject matter.