Shape-memory polymers (SMPs) undergo significant macroscopic deformation upon the application of an external stimulus (e.g., heat, electricity, light, magnetism, moisture and even a change in pH value). They have been widely researched since the 1980s and are an example of a promising smart material. This paper aims to provide a comprehensive review of SMPs, encompassing a fundamental understanding of the shape-memory effect, fabrication, modeling and characterization of SMPs, various actuation methods and multifunctional properties of SMP composites, and potential applications for SMP structures. A definition of SMPs and their fundamentals are first presented. Next, a description of their fabrication, characterization and constitutive models of SMPs are introduced. SMP composites, which act to improve a certain function as functional materials or the general mechanical properties as structural materials, are briefly discussed. Specially, the SMP composites can be developed into multifunctional materials actuated by various methods, such as thermal-induced, electro-activated, light-induced, magnetic-actuated and solution-responsive SMPs. As smart materials, the applications of SMPs and their composites receive much interest, including deployable structures, morphing structures, biomaterials, smart textiles and fabrics, SMP foams, automobile actuators and self-healing composite systems.