The stability and breathing motions about finitely deformed states of cylindrical shells of arbitrary wall thickness subjected to uniform dead load traction are investigated. The material is assumed to be a polynomial compressible material which is homogeneous, isotropic and hyperelastic. The stability of the finitely deformed state and small, free, radial vibrations about this state are investigated using the theory of small deformations superposed on large elastic deformations. The loss of stability occurs when the motions cease to be periodic. Numerical results are provided to study the effect of several geometric and material parameter. (C) 1997 Elsevier Science Ltd.