In hot extrusion, knowledge of the temperature distribution and the extrusion power is very important for the determination of the optimal process conditions. In this paper, the combined upper-bound method and finite-element method (FEM) for the optimal design of axisymmetric dies is presented. The billet material during plastic deformation process is modelled as a visco-plastic rate-sensitive material. The material flow stress is considered to be strain-rate and temperature dependent. In the upper-bound analysis, a stream-function form which produces a kinematically admissible velocity field is assumed. The Galerkin FEM is used to solve the axisymmetric heat-conduction equation with convective terms to obtain the temperature distribution in the deformation zone. The combined upper bound method/FEM is applied to analyse four different die shapes, namely, stream-lined, cosine, hyperbolic and conical. From the study it is observed that the optimal die length decreases with friction factor and increases with reduction ratio and ram velocity. The extrusion power required is lowest for the stream-lined die with cosine die following closely behind. The hyperbolic die is, however, better than the conical die at lower reduction ratios, whilst the conical die is superior at greater reduction ratios.