The elastic-plastic buckling of cylindrical shells under torsion is analysed with a deep thick-shell model under various boundary conditions. The word 'deep' means that in the general equations of equilibrium the three non-linear terms that involve the torsional force are all retained for the buckling analysis. In the Donnell-type shallow-shell theory, however, only one of such terms is retained. The word 'thick' means that in calculating strains and stress resultants the factor (1+z/R) is retained. This factor results from the trapezoid-like shape of the cross-section and is usually neglected in the thin-shell theory. For boundary conditions, not only the conventional geometrical boundary conditions, which are in terms of displacements and rotations, but also the mechanical boundary conditions, which are in terms of forces and moments, are considered. The numerical results of examples assess the effect of the additional non-linear terms, the effect of the factor (1+z/R), and the effect of the mechanical boundary conditions.