A comprehensive geometrically nonlinear theory is developed for anisotropic laminated composite sandwich plates and shells with geometrical imperfections, featuring a transversely incompressible core. The theory incorporates the asymmetry of the face sheets as well as geometric imperfections. An elasticity-based approach is used, which includes the displacement field and the Green-Lagrange strain-displacement tensor. Constitutive equations are derived from generalized Hooke’s Law, and the equations of motion are formulated using Hamilton’s PrincipleHamilton’s Principle. Two forms of the governing equations are presented: one as a mixed formulation and the other as a displacement formulation. In these formulations, core properties are considered for both weak (soft) and strong core architectures. The governing equations are simplified for various structural configurations. Finally, a detailed discussion of different boundary condition types is provided.