In addition to rotation, non-circular cross sections of twisted rods undergo longitudinal displacement, which causes warping of the cross section. This warping is independent of the longitudinal z coordinate and is a harmonic function of the (x,y) coordinates within the cross section. The Prandtl stress function is introduced, in terms of which the shear stresses are given as its gradients. This automatically satisfies equilibrium equations, while the compatibility conditions require that the stress function is the solution to Poisson’s equation. From the boundary condition of a traction-free lateral surface, it follows that the stress function is constant along the boundary of the cross section. The applied torque is related to the angle of twist by the integral condition of moment equilibrium. This theory is applied to determine the stress and displacement components in twisted rods of elliptical, triangular, rectangular, semi-circular, grooved-circular, thin-walled open, thin-walled closed, and multicell cross sections. The expressions for the torsional stiffness are derived in each case. The maximum shear stress and the warping displacement are also evaluated and discussed.