The solution of the boundary-value problem for the region of the transverse cross section of a curved rod is investigated in relation to the function characterizing the warping of cross sections in elastic torsion. Analytical solutions are given for two problems involving the stressed-strained state of a rod whose axis is curved along a circular arc due to bending by a transverse force and torsion concentrated torque. It is shown that the deflection perpendicular to the plane of the initial curvature of the rod axis is accompanied by elastic rotation of the transverse cross sections. It is found that an additional stress system develops during the torsion of a curvilinear rod (in comparison with a straight rod). Some other characteristics of the stressed-strained state of curved rods are described.