Starting with the pioneering computations of Stokes in 1847, the search of traveling waves in fluid mechanics 

has always been a fundamental topic, since they can be seen as building blocks to determine the long time dynamics. 

In this talk we shall discuss the existence of time quasi-periodic traveling wave solutions for three-dimensional 

pure gravity water waves in finite depth, on flat tori, with an arbitrary number of speeds of propagation. 

These solutions are global in time, they do not reduce to stationary solutions in any moving reference frame 

and they are approximately given by finite sums of Stokes waves traveling with rationally independent speeds of propagation. 

The major difficulties arises from the fact that the 3D water waves system is a quasi-linear PDE in higher space dimension with ``weak dispersion relation’’. As a consequence, in the search for quasi-periodic solutions one must deal with the presence of very strong resonance phenomena. 

We will focus on  the spectral analysis of the linearized equations at any approximate traveling wave solutions. 

The strategy is suited for higher dimensional dispersive PDEs with sublinear dispersion.