  The present article is concerned with mirror symmetry for generalized K3
surfaces, with particular emphasis on complex and K\"ahler rigid structures.
Inspired by the works of Dolgachev, Aspinwall-Morrison and Huybrechts, we
introduce a formulation of mirror symmetry for generalized K3 surfaces by Mukai
lattice polarizations, fixing the problems in the conventional formulation of
mirror symmetry for K3 surfaces. In particular, we settle the long-standing
problem of mirror symmetry for singular K3 surfaces. Along the way, we
investigate complex and K\"ahler rigid structures of generalized K3 surfaces.
