Elastic stability is perhaps the oldest topic in finite elasticity theory. Stability in the sense of Lyapunov is essentially a dynamic concept. Dynamics of continuous media find their proper place under the wings of thermodynamics. The thermodynamic foundation of the theory of elastic stability is now fairly secure, thanks to Duhem and Ericksen. Some subtle mathematical difficulties are still connected with questions of Frechet differentiability of the elastic energy functional. Applications to specific problems of buckling of structures have preceded the complete development of the theory of elastic stability. This situation is indeed quite common in engineering science. The significance of post-buckling behaviour has been recognized much later. This essentially nonlinear aspect, so characteristic for finite elasticity theory, will be reviewed in some detail.