Morphing structures, defined as body panels that are capable of a drastic autonomous shape transformation, have gained importance in the aerospace, automotive, and soft robotics industries since they address the need to switch between shapes for optimal performance over the range of operation. Laminated composites are attractive for morphing because multiple laminae, each serving a specific function, can be combined to address multiple functional requirements such as shape transformation, structural integrity, safety, aerodynamic performance, and minimal actuation energy. This paper presents a review of laminated composite designs for morphing structures. The trends in morphing composites research are outlined and the literature on laminated composites is categorized based on deformation modes and multifunctional approaches. Materials commonly used in morphing structures are classified based on their properties. Composite designs for various morphing modes such as stretching, flexure, and folding are summarized and their performance is compared. Based on the literature, the laminae in an n-layered composite are classified based on function into three types: constraining, adaptive, and prestressed. A general analytical modeling framework is presented for composites comprising the three types of functional laminae. Modeling developments for each morphing mode and for actuation using smart material-based active layers are discussed. Results, presented for each deformation mode, indicate that the analytical modeling can not only provide insight into the structure's mechanics but also serve as a guide for geometric design and material selection.