Abstract An upper-bound analysis is proposed for the extrusion of square sections from square billets through curved dies having prescribed profiles. Kinematically-admissible velocity fields for the purpose are derived using the dual-stream-function technique. Analytical results are presented for both frictionless and sticking friction conditions; for the latter situation the die geometry has been optimised with respect to appropriate parameters. It is shown that a cosine-shaped die with zero entry and exit angles yields the lowest extrusion pressure in the absence of friction, whilst the best upper-bound is provided by a straight tapered die under sticking-friction conditions. The internal work of deformation, however, is still found to be minimum for a straight die for frictionless extrusion.