  Some novel numerical approaches to solving direct and inverse obstacle
scattering problems (IOSP) are presented. Scattering by finite obstacles and by
periodic structures is considered. The emphasis for solving direct scattering
problem is on the Modified Rayleigh Conjecture (MRC) method, recently
introduced and tested by the authors. This method is used numerically in
scattering by finite obstacles and by periodic structures. Numerical results it
produces are very encouraging. The support function method (SFM) for solving
the IOSP is described and tested in some examples. Analysis of the various
versions of linear sampling methods for solving IOSP is given and the
limitations of these methods are described.
