The stability of homogeneous, isotropic, compressible, hyperelastic, thick spherical shells subjected to external dead-load traction are investigated within the context of the finite elasticity theory. 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 frequencies of small free vibrations about the pre-stressed state are obtained numerically. The loss of stability occurs when the motions cease to be periodic. The critical values of stress and deformation are given for a foam rubber, slightly compressible rubber and a nearly incompressible rubber. (C) 2001 Academic Press.