This paper considers the normal form of non-linear control systems. First we propose a generalized relative degree (relative degree vector) for non-linear single (respectively, multiple) input control system, which is called the point relative degree (respectively, point relative degree vector). For the systems without output, the concepts of essential relative degree (respectively, essential relative degree vector) and the essential point relative degree (respectively, essential point relative degree vector) are defined. Unlike the classical definition which requires regularity, the point relative degree (vector) is always well defined. Using these new concepts the generalized normal form is obtained. Its relationship with the Jacobian linearization is investigated. Using it, a straightforward computation algorithm is provided to achieve the generalized normal form. Based on the generalized normal form we prove that with an additional condition, if the zero-dynamics is stable the overall system is stabilizable by using pseudo-linear state feedback control. For the systems under generalized normal form with unstable zero dynamics, the centre manifold approach is applied. It is shown that the stabilization technique via a designed centre manifold is still applicable to this kind of general non-linear control system.