This paper is concerned with the elastic flexural buckling of structural members under torsion, and with second-order moments in torsion members. Previous research is reviewed, and the energy method of predicting linear buckling is presented. This is used to develop the differential equilibrium equations for a buckled member. Approximate solutions based on the energy method are obtained for a range of conservative applied torque distributions and flexural boundary conditions. A comparison with the limited range of independent solutions available and with independent finite element solutions suggests that the errors in the approximate solutions may be as small as 1%. The predicted linear elastic buckling torques may be used to approximate the second-order bending moments caused by torsion in members under more general loading. A method is developed for approximating these second-order moments. This is used as the basis of a method of estimating when these second-order moments may be significant by comparing the actual member slenderness with a reference value. Reference values of slenderness are calculated for two examples involving an equal angle member and a circular hollow section member (both simply supported), and the importance of second-order torsion effects in an I-section member is estimated. The reference values of slenderness are found to be very high, and it is concluded that second-order moments caused by torsion in typical structural steel members with slenderness ratios L/ry less than 300 are very small and may be neglected.