"The study of Arnold diffusion in Hamiltonian systems has received a lot of attention in the last years. This phenomenon arises when a small perturbation in a system causes big changes in it leading to global instabilities. In recent years there have been partial results characterizing this phenomenon and proving its existence in Hamiltonian systems but mostly for systems of low dimension. This project wants to be a step forward showing that such instabilities also arise in Dynamical Systems on lattices and in Hamiltonian Partial Differential Equations (HPDEs), which can be seen as Dynamical Systems of infinite dimension. We will focus our attention on two different problems: the Fermi-Pasta-Ulam model and the energy transfer phenomenon in Hamiltonian PDEs.Regarding the first part, we will consider the Fermi-Pasta-Ulam model (FPU), which is a model of a discretized nonlinear string. One would expect that as time evolves, the system reaches equipartition of energy among the modes. Nevertheless, numerical experiments by Fermi, Pasta and Ulam (1955) showed that in some settings it is not reached in the time range for which the numerical experiments are reliable. This fact is called the FPU paradox. To understand how the equipartition of energy can be reached after longer times we plan to find instability mechanisms in the low energy regime.In the second part we will prove the existence of solutions of some HPDEs in the d dimensional torus which undergo transfer of energy to higher modes as time tends to infinity. This transfer of energy can be measured by the growth of high Sobolev norms. First, we plan to prove the existence of orbits with arbitrarily large finite growth of Sobolev norms for different Hamiltonian PDEs. Finally we plan to prove Bourgain's conjecture, which asserts the existence of orbits of the cubic defocusing nonlinear Schrodinger equation in the two torus whose s-Sobolev norms tend to infinity as time tends to infinity."