The previous chapter gave a rather detailed description of bifurcations of equilibria and fixed points in generic one-parameter families of ODEs and maps with minimal possible dimension of the state space. These results are also applicable to general n-dimensional systems because of the existence of a low-dimensional invariant manifold for parameter values near the bifurcation point, on which all interesting local dynamics in the state space is concentrated. The present chapter is devoted to the constructive definition of this invariant Centre Manifold and the efficient computation of normal forms on it yielding all the relevant information. Our proof will also establish the existence and smoothness of the local stable and unstable invariant manifolds of hyperbolic saddles.