This paper contains some results on topological stability (see [2, 31) that generalize those obtained in [2] much in the same way as Lyapunov’s direct theorem generalizes the asymptotic stability results of the hyperbolic case: if at a critical point, the linear part of a vector field has proper values with negative real parts, the point is asymptotically stable and the vector field has a quadratic Lyapunov function; however, asymptotic stability may also be proved for vector fields with non-hyperbolic linear approximations, provided they have a Lyapunov function. In a way, this is what we do here, letting Anosov diffeomorphisms play the role of the hyperbolic critical point and replacing stability by topological stability; we get this time a class of topologically stable diffeomorphisms wider than the class of Anosov diffeomorphisms. The same approach-that combines Lyapunov functions with some of the ideas of [4] -may also be applied to obtain a similar generalization for Axiom A diffeomorphisms. Theorem 2.1 below states essentially that a diffeomorphism f of a compact riemannian manifold M is Anosov if and only if its tangent map has a nondegenerate Lyapunov quadratic function. From this quadratic form we may construct easily a Lyapunov function for f, i.e., a real function I’ defined on a neighbourhood of the diagonal in M x M such that V(x, X) = 0, x E M, and