Like the human limbs which inspired them most robots are discrete mechanisms with rigid links connected by single degree of freedom joints. In contrast, 'continuum' and 'serpentine' robot mechanisms move by bending through a series of continuous arcs producing motion which resembles that of biological tentacles or snakes. This paper provides a single reference to the expanding technology of continuum robot mechanisms. It defines the fundamental difference between discrete, serpentine and continuum robot devices, presents the 'state of the art' of continuum robots, outlines their areas of application, and introduces some control issues. Finally, some conclusions regarding the continued development of these devices are made.