The averaging method is very famous in non-linear vibration. This chapter describes it from the viewpoint of geometry and Poincare mapping. We will introduce it according to the theory of KBM (Krylov, Bogoliubov, Mytropolisky). This method is very effective for the weak non-linear system and the linear disturbed system [1]. We shall prove that in certain condition the use of the averaging method enables us to obtain information on the global dynamical behaviour of the semi-infinite time domain. The theory of the averaging method has various forms. When we describe the averaging method in the theory of bifurcation, we base our statements on the work of KBM and Hale (J.K. Hale) [30].