A theory is outlined of sandwich box-type composite shells designed to withstand a combination of thermal loading, internal pressure, torsional and axial loads. A cross section of the shell represents a rectangular box with curved cylindrical sections at the corners. The facings of the shell are dissimilar to maximize their efficiency, according to the loads acting on each facing. This approach enables a designer to optimize the structure by maximizing the load-carrying capacity or minimizing the weight. The formulation includes the following developments: 1) global theory of a sandwich shell composed of rectangular and cylindrical sections, where equations of motion are formulated based on a first-order shear deformable version of Sanders's shell theory; 2) theory for local deformations and stresses in the facings, where the facing is treated as a thin geometrically nonlinear plate or shell on an elastic foundation using von Karman's approach and the elastic foundation represents a support provided by the opposite facing and the core; and 3) an outline of an enhanced micromechanical constitutive formulation based on the incorporation of the effect of the thermomechanical coupling on the material properties and temperature.