Chapter 1 deals with basic properties of the category of quasi-uniform spaces and its full subcategory Qut of transitive quasi-uniform spaces. Chapter 2 concerns Fletcher's construction. We extend the class of covers to which this construction may be applied and study the functoriality of the construction. The major result is that every right inverse of the forgetful functor Qut--->Top is obtainable by the extended Fletcher construction. In Chapter 3 we characterize pairwise zero dimensional bitopological spaces as those admitting transitive quasi-uniformities. An initiality characterization of pairwise zero dimensional bitopological spaces suggested by Brümmer leads to a description of the coarsest right inverse of the forgetful functor. In Chapter 4 we discuss countably based transitive quasi-uniformities, in that they relate to quasi-metrization. We elaborate on a result of Fletcher and Lindgren (1972) and obtain a bitopological analogue. In Chapter 5 we bring together a number of topics which relate to our previous chapters and point to further questions.