Symplectic numerical integration theory for Hamiltonian systems has been developed rapidly in past years.The recent monographs [1] and [2] summarize the main developments and important results of this theory.Qualitative behavior of symplectic integrators applied to Hamiltonian systems has been investigated by many authors.Some stability results either in the spirits of the KAM theory or based on the backward analysis have been well established.The typical stable dynamics of Hamiltonian systems,e.g.,quasi-periodic motions and their limit sets - minimal invariant tori,can be topologically preserved and quantitatively approximated by symplectic integrators.In this talk I give a brief review about the stability results.