A numerical study is carried out on the conjugate thermal transport in polymer and food melts flowing through extrusion dies. The simulation is performed to determine the influence of conduction through the die wall and of the thermal boundary conditions on the transport in the fluid and on the conditions at the outlet. An extrusion die with a uniform temperature or heat transfer coefficient specified at the outer surface is considered. It is found that, because of conduction in the solid wall, important physical variables such as centerline velocity, pressure drop, bulk temperature of the fluid and shear experienced by the fluid are strongly affected by the boundary conditions, as well as by the wall thermal conductivity and thickness. Channels of different geometries are used for the study. The flow in a circular straight tube with constant wall thickness is studied first. Flow and thermal transport in different, constricted, channels are studied next, Different wall materials are also considered. Comparisons with some experimental results are presented, indicating good agreement. The fluids considered In this study are highly viscous, polymer melts. Due to high viscous dissipation and temperature-dependent viscosity, the flow and heat transfer are coupled and the problem is quite complicated. The results show that, for some operating conditions, the bulk temperature can be high enough to cause significant heat transfer from the fluid to the wall. The downstream variation in the pressure and temperature are calculated. The thermal boundary conditions are found to have a strong influence on the temperature field and thus on the flow. The general dependence of pressure drop on temperature, flow rate, and geometry is investigated. Several other basic aspects of this problem are also discussed.