Local and Semi Local Bifurcations in Hamiltonian Dynamical Systems Bifurcations of Equilibria Bifurcations of Periodic Orbits Bifurcations of Invariant Tori Perturbations of Ramified Torus Bundles Planar Singularities Stratifications Normal Form Theory Proof of the Main KAM Theorem Proofs of the Necessary Lemmata. Once again KAM theory is committed in the context of nearly integrable Hamiltonian systems. While elliptic and hyperbolic tori determine the distribution of maximal invariant tori they themselves form n parameter families. Hence without the need for untypical conditions or external parameters torus bifurcations of high co dimension may be found in a single given Hamiltonian system. The text moves gradually from the integrable case in which symmetries allow for reduction to bifurcating equilibria to non integrability where smooth parametrisations have to be replaced by Cantor sets. Planar singularities and their versal unfoldings are an important ingredient that helps to explain the underlying dynamics in a transparent way.