The talk illustrates some recent results on the stability properties of elliptic equilibria of Hamiltonian systems. The employed approach is based on the methods of Hamiltonian perturbation theory, specifically, on Nekhoroshev's theory, which is capable of giving stability results for finite, but extremely long, times. The interest of this kind of `practical stability' is in that it applies to equilibria which are Lyapunov unstable, but nevertheless, their instability manifests itself only on extremely long time scales: for practical purposes, they may be considered as stable.

From the technical point of view, the key notion is that of `directional quasi--convexity' of the fourth order Birkhoff normal form, which extends the `quasi--convexity' hypothesis of the standard Nekhoroshev's theory and appears to be a rather natural ans easily checkable hypothesis for the problem. Applications to two problems from Celestial Mechanics are described, namely, the stability of the equilateral Lagrangian of the restricted spatial three body problem and that of the Riemann ellipsoids.