Developments in surgery have been geared toward minimizing the invasiveness of the procedure to improve both the treatment itself and the patient's postoperative wellbeing. As such, attention has been directed toward reducing human error and miniaturizing clinical devices by developing smaller devices and robotic systems. While there have already been significant advancements in this area, apparatus can further benefit from being foldable, expandable, and further condensable. By promoting these characteristics, origami engineering, which extrapolates the fundamental principles of paper folding to real-world projects, has become increasingly prevalent in the biomedical field. This paper reviews the field of origami engineering, its fundamental mechanical and mathematical properties, and the recent progress in specific research areas. Then, this paper discusses several devices that have emerged over the past decade in detail based on their characteristics and implementations. Finally, this paper addresses the technical challenges and general research trend of self-folding non-invasive miniature robots.