In this thesis we examine the integrability of Hamiltonian normal form and the perturbations of superintegrable system. The integrals of a Hamiltonian system are functions that are fixed along the solutions. In fact, the solutions of a Hamiltonian system are located on the level curves of the integrals of that system. Therefore, if a Hamiltonian system with n degree of freedom has n functionally independent integral in the involution In this case, the system is integrable and its solutions can be obtained. A dynamical system with n degree of freedom that the number of its integral of motion is greater than n, is an superintegrable system. Here, we first examine the integrability of the normal forms of Hamiltonian systems with two and three degrees of freedom with some first order resonance and we discuss two integrable normal form Hamiltonian chains, FPU and ?:?:?:?:?:?, and three non-integrable normal form Hamiltonian chains, with emphasis on the ?:?:?:?:?:? resonance. In addition, the time series H_? (t) and the integral diagram H_? in the appropriate action variable are introduced as a predictor of integrability and used to determine the integrability of a system. In the end, we study geometry of the dynamics of the superintegrable systems and study how to change them under the perturbations. .