The aim of this work is the study of symmetric linear multistep methods applied to Hamiltonian systems; we show that this class of methods can have good properties of near preservation of the energy and momenta for long-time integrations of Hamiltonian systems. In Chapter 1 and 2 we study partitioned linear multistep methods applied to first order Hamiltonian equations, and we show how the use of symmetric partitioned multistep method can lead to near preservation of energy for a specific class of separable Hamiltonians. In Chapter 3, 4 and 5 we study symmetric linear multistep methods applied to second order constrained Hamiltonian systems. In Chapter 3 and 4 we focus on the theoretical analysis of the excellent behaviour that this class of methods presents on this kind of problems; we show as well the construction of these methods, and some numerical experiments. In Chapter 5 we study the optimization of the implementation of this class of methods.