Summary: This paper deals with thin-walled beams where the cross-section varies along the length. The beams are assumed to be fabricated from an arbitrary number of flat plates (in general tapered) joined along their edges, so that a cross-section consists of a set of connected thin rectangles. Using a variational approach to derive the Euler-Lagrange equations for the displacement components, the equilibrium equations defining instability phenomena, and the corresponding limit conditions are obtained for thin, open cross-section, continually variable beams. The variability of the cross-section introduces an additional term to the expression of the tangential tensions modifying the shear centre position. Therefore, on the basis of the positive definiteness of the second variation of the total potential energy, a stability criterion is presented. In fact the Euler-Lagrange equations can be used to study the instability of such beams. The whole analysis lies within the constraining assumption of linear and elastic behavior.