We shall discuss the complex dynamics of Hamiltonian or reversible PDEs, like the NLS, the Klein-Gordon equation, the wave equation in several dimension,the water waves equations of hydrodynamics, focusing on both chatotic and regular behaviour as the existence of quasi-periodic solutions.We shall discuss also long time existence results obtained by Birkhoff normal forms. We shall develop general tools of nonlinear analysis, like the Nash-Moser implicit function theorem and the theory of pseudo-differential operators.The first part of the course will present general tools of pseudo differential and para-differential operator theory.