The nonlinear Schrodinger and Klein-Gordon equations appear as good models for many physical phenomena in nonlinear optics or plasma physics. The mathematical study of their Cauchy problems depends on the space dimension, the growth of the nonlinearity and the regularity of the initial data.

In this talk, first I will define an energy critical model for the nonlinear Klein-Gordon and Schrodinger equations in two space dimensions. Then, I will explain why one has global well-posedness in the so-called sub-critical and critical cases, while a sort of ill-posedness occurs in the super-critical regime. Finally, I will state the scattering result for the Klein-Gordon case and compare it to the power case.

I will conclude my talk by some remarks about the focusing case.