Abstract: Unfolding and folding problem is a popular research topic in computer graphics, and has a wide range of applications, such as industrial manufacturing, architectural design, medical treatment, and aviation technology. In this survey, we review the basic concepts of unfolding and folding problem, introduce the research and application in four fields: robot design, computer animation, deep learning and others. We discuss the research work of unfolding and folding problem in detail. First, according to the different degrees of unfolding, we summarize research progress and typical algorithm ideas from two aspects: full unfolding and approximate unfolding. Full unfolding is to unfold 3D objects into 2D space without overlapping and deformation. However, most objects cannot be directly unfolded, and only an approximately unfolded structure can be solved. Approximate unfolding is a non-overlapping and deformed process, which is unfolded into the plane domain by mapping. How to find the smallest deformation is the key to approximate unfolding. Second, according to the different folding forms, the folding problem is divided into two types: Origami and Kirigami. We divide Origami into rigid folding and curved folding according to the different forms of crease, such as straight crease and curved crease. Kirigami is a special folding method that combines cutting and folding technology, which drives folding by the elastic force or other external forces generated by cutting. Here, we mainly consider the technology or algorithm of using Kirigami technology to construct auxetic structures. In addition, in order to compare the advantages and disadvantages of the algorithm, we summarize the commonly used algorithm indicators of unfolding and folding algorithm. Then, we evaluate the typical algorithm in recent years, and analyze advantages and disadvantages. Finally, we summarize and propose the development trend of unfolding and folding, including algorithm accuracy and robustness, fold volumetric objects, self-driven process and intelligent application of Kirigami technology.