  We review some recent results on nonlinear Schrodinger equations with
potential, with emphasis on the case where the potential is a second order
polynomial, for which the interaction between the linear dynamics caused by the
potential, and the nonlinear effects, can be described quite precisely. This
includes semi-classical regimes, as well as finite time blow-up and scattering
issues. We present the tools used for these problems, as well as their
limitations, and outline the arguments of the proofs.
