  Global existence and scattering for the nonlinear defocusing Schr\"odinger
equation in 2 dimensions are known for domains exterior to star-shaped
obstacles and for nonlinearities that grow at least as the quintic power. In
this paper, we extend the global existence result for all non-trapping
obstacles and for nonlinearities with power strictly greater than quartic. For
such nonlinearities, we also prove scattering for a class of so-called almost
star-shaped obstacles.
