The theory is based on an assumed displacement field, in which the surface displacements are expanded in powers of the thickness coordinate up to the third order. The theory allows parabolic description of the transverse shear stresses, and therefore the shear correction factors of the usual shear deformation theory are not required in the present theory. The theory accounts for small strains but moderately large displacements (i.e., von Karman strains). Exact solutions for certain cross-ply shells and finite-element models of the theory are also developed. The finite-element model is based on independent approximations of the displacements and bending moments (i.e., mixed formulation), and therefore only C°-approximations are required. Further, the mixed variational formulations developed herein suggest that the bending moments can be interpolated using discontinuous approximations (across inter-element boundaries). The finite element is used to analyze cross-ply and angle-ply laminated shells for bending, vibration, and transient response. Numerical results are presented to show the effects of boundary conditions, lamination scheme (i.e., bending-stretching coupling and material anisotropy) shear deformation, and geometric nonlinearity on deflections and frequencies. Many of the numerical results presented here for laminated shells should serve as references for future investigations. / Ph. D.