We study the theory of Scattering in the energy space for various nonlinear Schr?dinger equations. In dimension 3 or bigger we consider a variable coefficients equation, for a gauge invariant, defocusing nonlinearity of power type on an exterior domain with Dirichlet boundary conditions. In order to prove scattering, we prove a bilinear smoothing (interaction Morawetz) estimate for the solution and, under the conditional assumption that Strichartz estimates are valid for the linear flow, we prove global well posedness in the energy space for energy subcritical powers, and scattering provided the power is mass supercritical. When the domain is the whole space, by extending the Strichartz estimates due to Tataru, we prove that the conditional assumption is satisfied and deduce well posedness and scattering in the energy space. In low dimension spaces of dimension 1,2 or 3, we simplify the scattering theory in Rn for the Schrödinger equation, generalizing it to the system framework. Joint work with Piero D'Ancona and Mirko Tarulli.