Walking robots have long been proposed as solutions to the problem of mobile machines operating in unstructured and natural environments because they can traverse relatively large obstacles and avoid dangerous or sensitive areas of the ground. Walking machines that can move in any direction and turn through any radius have an 'omnidirectional manoeuvring' capability. Hexapodal (six-legged) walking machines have advantages over bipedal and quadrupedal machines in the areas of stability, payload and simplicity, as well as the ability to operate with damaged or broken legs. Walking robots inspired by insect neurology and biology continue to extend the state of the art, but using this 'biologically inspired' approach to develop behaviours not exhibited by the animals requires further insight into the principles of walking. Part of this thesis develops a possible explanation for the distributed inter-leg influences that control walking in the stick insect: that legs swing forward to take their next step when the proximity of other legs or workspace boundaries restricts their movement. This new concept, called the 'restrictedness' algorithm, has similarities to distributed gait controllers in the literature but with enhanced capabilities and simpler implementation. The restrictedness algorithm is a significant contribution to the field of walking robotics and may also be of relevance to future biological studies. Omnidirectional walking requires the adaptation of leg movements to the locomotion direction and robustness to unusual leg lifting positions. A novel velocity field specified in a 'task coordinate system' for each leg is described, a scheme that allows simple realtime adaptation to different commanded body movement vectors and changing spatial relationships between the legs. Active compliance and automatic levelling improve the robot's interaction with the ground, especially over rough ground. Experiments with the robot Hamlet, designed and built primarily by the author, proved that the new restrictedness and velocity field swing algorithms display the important traits required from a walking robot gait controller. Specifically, the algorithm provided delivers the range of optimal forward walking gaits observed in insects while also supporting omnidirectional walking, including rotating while simultaneously translating. Optimization by simulating a model of the robot improved the controller's forward walking performance.