In this paper, the flexural-torsional buckling and free vibration of tapered thin-walled beam-columns with arbitrary cross-section shape are extensively investigated. The governing equilibrium equations and motion equations are obtained from the stationary condition of the potential energy. The strain energy is derived in presence of initial stresses. In the work of the applied forces, effects of load eccentricities from shear and centroid centerlines are taken into account. Free vibration is considered in presence of harmonic excitations. In presence of arbitrary boundary conditions and variable cross-section properties, a semi-analytical approach based on power series method is adopted in solution. According to this method, displacements and geometric constants are approximated by polynomial functions up to a certain order, where accurate results are reached. The flexural-torsional buckling loads or natural frequencies are determined by solving an eigenvalue problem. In order to measure the accuracy and to check the validity of the present method, several examples including flexural-torsional behavior and free vibration analysis of non-prismatic thin-walled members with web and flange tapering and various boundary conditions are considered. The obtained results are compared to the finite element simulations and other available solutions.