This study presents different analytical and finite element models for sandwich structures with functionally graded core. The trade-off between weight and stiffness as well as a comparison between these structures and sandwich structures with homogenous core is also presented. The problem of low-velocity impact between a sandwich structure with functionally graded core and a rigid spherical indentor is solved. A few advantages and disadvantages of these types of structure are presented. Although sandwich structure offer advantages over other types of structures, it is important to develop new types of materials in order to obtain the absolute minimum weight for given conditions (e.g., structural geometry and loadings). One alternative is represented by functionally graded materials (defined as materials with properties that vary with location within the material in order to optimize a prescribed function). With the new developments in manufacturing methods these materials can be used for a large number of applications ranging from implant teeth to rocket frames