Origami-inspired systems are attractive for creating structures and devices with tunable properties, multiple functionalities, high-ratio packaging capabilities, easy fabrication, and many other advantageous properties. Over the past decades, the community has developed a variety of simulation techniques to analyze the kinematic motions, mechanical properties, and multiphysics characteristics of origami systems. These various simulation techniques are formulated with different assumptions and are often tailored to specific origami designs. Thus, it is valuable to systematically review the state-of-the-art in origami simulation techniques. This review presents the formulations of different origami simulations, discusses their strengths and weaknesses, and identifies the potential application scenarios of different simulation techniques. The material presented in this work aims to help origami researchers better appreciate the formulations and underlying assumptions within different origami simulation techniques, and thereby enable the selection and development of appropriate origami simulations. Finally, we look ahead at future challenges in the field of origami simulation.