There has been a boost of research activities in robotics using soft materials in the past ten years. It is expected that the use and control of soft materials can help realize robotic systems that are safer, cheaper, and more adaptable than the level that the conventional rigid-material robots can achieve. Contrary to a number of existing review and position papers on soft-material robotics, which mostly present case studies and/or discuss trends and challenges, the review focuses on the fundamentals of the research field. First, it gives a definition of softmaterial robotics and introduces its history, which dates back to the late 1970s. Second, it provides characterization of soft-materials, actuators and sensing elements. Third, it presents two general approaches to mathematical modelling of kinematics of soft-material robots; that is, piecewise constant curvature approximation and variable curvature approach, as well as their related statics and dynamics. Fourth, it summarizes control methods that have been used for soft-material robots and other continuum robots in both model-based fashion and model-free fashion. Lastly, applications or potential usage of soft-material robots are described related to wearable robots, medical robots, grasping and manipulation.