Fully developed flow of an incompressible Newtonian fluid through a duct in which the orientation of the cross section is twisted about an axis parallel to an imposed pressure gradient is analyzed here with the aid of the penalty/Galerkin/finite element method. When the axis of twist is located within the duct, flow approaches limits at low and high torsion, the spatial frequency &tgr; by which the duct is twisted. For small torsion, flow is nearly rectilinear and solutions approach previous asymptotic results for an elliptical cross section. For large torsion, flow exhibits an internal layer structure: a rotating circular‐cylinder core with a nearly parabolic axial velocity profile, an internal layer of thickness &tgr;−1along the perimeter of the largest circular cylinder that can be inscribed in the duct, and nearly quiescent flow outside of the circular cylinder. The maximum rate of swirl in the core of a square duct is found to be at moderate torsion. The primary effect of inertia is an increase in pressure with distance from the axis, due to centrifugal acceleration. When the duct is offset from the axis of twist, inertia leads to one, two, or three primary vortices without apparent bifurcation of steady states, although stability of steady flows is lost beyond detected Hopf points.