The work contained in this thesis is concerned with the flow behaviour of polymer solutions and melts. It is a theoretical study, though it refers extensively to experimental work by others. Calculations are analytic where possible, though some numerical work is required. The principal source is the theoretical model of linear polymer melts developed by Doi and Edwards. The presentation is as follows: Chapter 1 reviews the experimental properties of polymeric liquids and some of the phenonenological schemes used to describe them. Review material extends into chapter 2 which covers molecular theory, the Doi-Edwards model, and developments in the last few years. Chapter 3 applies the model to the 'spurt effect', a well known unstable flow of linear polymer melts. We show that this may be explained quantitatively by the multivalued constitutive behaviour of the model. Chapter 4 examines the stability of the flow field associated with the spurt effect and shows how certain pulsing periodicities in the flow may be related to the 'normal stress' behaviour of the fluid. Chapter 5 is a case study of a more complex branched polymer: the 'H-polymer'. It is shown that both the linear and non-linear viscoelastic response may be calculated in a straightforward extension of the Doi-Edwards model, comparing favourably with experiments to date.