Abstract: The numerical methods of the equations of multibodysystem dynamics are hot topics in mathematics and mechanics.Especially, the differential-algebraic equations of multibody system dynamics usually are so-called the questions of index three, whose solutions remain very difficult. The popular approachs have their limitations, mainly relatd to the problemsof stiffness, and of drift. The symplectic integrators of the equations of multibody system dynamics on manifold, are new numerical schemes devoloped in recent years. The constraints and ordinary differential equations may be properly handled with symplectic methods, which are shown to be promising and efficient on the problems of drift and stiffness. In this paper we discuss the new theory and propose some questions to be solved.