Summary: A geometrically nonlinear theory of initially imperfect doubly curved sandwich shells is developed. The theory encompasses sandwich structures with dissimilar face sheets constructed of laminate composites and with the core modeled as an orthotropic body featuring strong or soft behaviors. The theory includes the dynamic as well as the temperature and moisture effects. As a special case, a range of related applications of the theory of sandwich shells with soft core are presented. These involve the buckling and postbuckling of flat and circular cylindrical shells compressed by uniaxially edge loads, and the influence played in this respect by a number of geometrical and physical parameters is highlighted. Also contained are comparisons of buckling results obtained in the context of the present theoretical model with the ones obtained experimentally, and reasonable agreements are reported.