This talk is devoted to introduce direct and inverse wave scattering problems arising in the context of linear elasticity for diffraction gratings (periodic structures). The mathematical topics to be discussed include: uniqueness and existence of weak solutions for the forward scattering via variational methods; numerical solutions using integral equation methods; uniqueness in identifying unknown scattering interfaces from near-field measurements; efficient inversion algorithm based on optimization scheme. These problems play an import role in industrial applications such as non-destructive testing for damage detection, investigation of earthquakes and oil bodies, etc.