KAM-theory for PDE with one and many space variables

In late 1980's a version of the classical KAM-theory, applicable to hamiltonian PDE with one space variable, was developed. Recently this theory was extended to some space-multidimensional hamiltonian PDE. Namely, in [1-3] two different approaches to treat the multidimensional equations were suggested, which were later developed by a number of researches. In my talk I will discuss the space-multidimensional "KAM for PDE" theory and its differences with the one-dimensional case.