A mathematical model is set up for melt flow and heat transfer in a slit die, coupled with heat transfer in the die body. The model is reduced to two dimensions by neglecting variations across the width of the die. Conduction, convection and viscous dissipation in the non-Newtonian melt are included, together with conduction and electrical heating in the die body and heat losses by free convection and radiation from the die surfaces. The model is implemented using finite elements, and provides detailed temperature fields on the die cross-section, displaying, for the first time, the effects of thermal coupling between the melt and die body. The consequences for processing are discussed for several situations, including horizontal and vertical operation of the die, with a polished or corroded surface, and with or without insulation. The appropriate use of dimensionless groups for the interpretation of the numerical results is discussed. A criterion in terms of a Pearson number is developed, for assessing whether an isothermal melt flow analysis is applicable. A simple analytic result is proposed for calculating the die set temperature required to minimise melt temperature variations. This is tested against the numerical results and is broadly validated.