Shape-memory polymers (SMPs) offer a number of potential technical advantages that surpass other shape-memory materials such as shape-memory metallic alloys and shape-memory ceramics. The advantages include high recoverable strain (up to 400%), low density, ease of processing and the ability to tailor the recovery temperature, programmable and controllable recovery behavior, and more importantly, low cost. This article presents the state-of-the-art regarding SMPs. First, the architecture, type, and main properties of the traditional and recently developed SMPs are introduced. Second, structural and multifunctional SMP composites are summarized and discussed. These composites greatly enhance the performance of the SMPs and widen their potential applications. Finally, current applications of SMP materials in aerospace engineering, textiles, automobiles, and medicine are presented.