Origami robots are created using folding processes, which provide a simple approach to fabricating a wide range of robot morphologies. Inspired by biological systems, engineers have started to explore origami folding in combination with smart material actuators to enable intrinsic actuation as a means to decouple design from fabrication complexity. The built-in crease structure of origami bodies has the potential to yield compliance and exhibit many soft body properties. Conventional fabrication of robots is generally a bottom-up assembly process with multiple low-level steps for creating subsystems that include manual operations and often multiple iterations. By contrast, natural systems achieve elegant designs and complex functionalities using top-down parallel transformation approaches such as folding. Folding in nature creates a wide spectrum of complex morpho-functional structures such as proteins and intestines and enables the development of structures such as flowers, leaves and insect wings. Inspired by nature, engineers have started to explore folding powered by embedded smart material actuators to create origami robots. The design and fabrication of origami robots exploits top-down, parallel transformation approaches to achieve elegant designs and complex functionalities. In this Review, we first introduce the concept of origami robotics and then highlight advances in design principles, fabrication methods, actuation, smart materials and control algorithms. Applications of origami robots for a variety of devices are investigated, and future directions of the field are discussed, examining both challenges and opportunities.