Abstract: Origami is to fold a two dimensional paper into the three dimensional structure without cutting and adhesion. With the merits of simple design, rapid forming and wide range of applications, origami has the promising applications in the fields of deployable structures, structural assembly and self-forming. Firstly, this paper reviews several typical origami applications, such as buckling-induced microscale three dimensional structures, foldable solar panels and DNA spiral assembled structures; Then, we define the classifications of origami according to different criteria, such as the number of curved creases, relative motion, the assumption of rigid folding surface, the number of used papers. Since the crease design problem is the key issue of origami, we focus on the origami crease design, including summarizing the basic principles of the crease design, addressing several typical crease design samples such as Miura, waterbomb, Yoshimura and diagonal crease designs. Furthermore, we introduce the distinctive features and geometrical relations of the typical crease design. For the recent innovative crease design methods, the improvement of the classic crease design, establishment of the crease design database, use of topology optimization method and the recent crease design algorithms are briefly discussed. Finally, we prospect the future research orientations of origami based on current research progress of origami, including the transformable structures, four-dimension origami, multi-material origami and multi-scale origami.