Large, fast, digital computers have been widely used in engineering practice and their use has had a large impact in many fields. Polymer processing is no exception, and there is already a substantial amount of literature describing ways in which processes can be analysed, designed or controlled using the potentialities of modern computers. The emphasis given varies with the application, and most authors tend to quote the results of their calculations rather than describing in any detail the way the calculations were undertaken or the difficulties experienced in carrying them out. We aim to give here as useful and connected an account as we can of a wide class of applications, for the benefit of scientists and engineers who find themselves working on polymer processing problems and feel the need to undertake such calculations. The major application we have in mind is the simulation of the dynamics ofthe various physical phenomena which arise in a polymer process treated as a complex engineering system. This requires that the system be reasonably well represented by a limited number of relatively simple subprocesses whose connections can be clearly identified, that the domiΒ nant physical effects relevant to each subprocess can be well defined in a suitable mathematical form and that the sets of equations and boundary conditions developed to describe the whole system can be successfully discretised and solved numerically.