Snake robots have advantages of terrain adaptability over wheeled mobile robots and traditional articulated robot arms because of their limbless thin body structure and high flexibility. They have extensive applications in tasks such as rescue, disaster recovery, inspection and minimally invasive surgery. Current research on snake robots is mainly focused on snake-like locomotion and the embodiment of these motion gaits for different applications. Modular structure and real-time control algorithms are two key aspects for snake robots operating in constrained environments. This review will attempt to address both. First, a review on the snake motion and the body structure is provided, which outlines the biological foundation of all snake robots. This is followed by the mechanical structure of snake robots, especially the structure of elemental snake modules. Finally, control algorithms for variant terrain contours and obstacle avoidance are discussed. The review also outlines emerging application areas and potential future directions of snake robots.