KAM and normal form methods are very powerful tools for analyzing the dynamics of nearly integrable finite dimensional Hamiltoniansystems. In the last decades, the extension of these methods to infinite dimensional systems, like Hamiltonian PDEs (partial differentialequations), has attracted the interest of many outstanding mathematicians like Bourgain, Craig, Kuksin, Wayne and many others. Thesetechniques provide some tools for describing the phase space of nearly integrable PDEs. More precisely they give a way to constructspecial global solutions (like periodic and quasi-periodic solutions) and to analyze stability issues close to equilibria or close to specialsolutions (like solitons). In the last seven years, I developed new methods for proving the existence of quasi-periodic solutions of quasi-linear, one-dimensional PDEs. This is an important step towards treating many of the fundamental equations from physics since mostof these equations are quasi-linear. In particular, this is the case for the equations in fluid dynamics, the water waves equation being aprominent example. These novel techniques are based on a combination of pseudo-differential and para-differential calculus, with theclassical perturbative techniques and they allowed to make significant advances of the KAM and normal form theory for one-dimensionalPDEs. On the other hand, many challenging problems remain open and the purpose of this proposal is to investigate some of them. Themain goal of this project is to develop KAM and normal form methods for PDEs in higher space dimension, with a particular focus onequations arising from fluid dynamics, like Euler, Navier-Stokes and water waves equations. By extending the novel approach, developedfor PDEs in one space dimension, I have already obtained some preliminary results on PDEs in higher space dimension (like the Eulerequation in 3d), which makes me confident that the proposed project is feasible.