The breakthrough of Mouhot and Villani (2011) on the so called Landau Damping for the collision-less Vlasov equation \begin{eqnarray} & \partial_t F +v\cdot
abla_x F \pm
abla_x V\cdot
abla_v F=0 ; \hbox{with} ; (x,v) \in (\Omega \subset (\mathbf R_v)^d\times \mathbf R^d)\\ & -\Delta V =G+f = F- \langle F\rangle, ; \langle F \rangle =\int_{\Omega} F(x,v,t) dx \end{eqnarray} has generated in the mathematical community many activities around the qualitative behavior (in particular for large time) of their solutions and for the behavior of the space independent average: \begin{equation} G(v,t) = \int_{\Omega} G(t,v,x)dx,. \end{equation} As I intend to show, starting from physical motivation, such behavior depends on the spectra of the linearized operator and on the size of the perturbation. Existing mathematical results are in line with a very rich and diverse behavior.