Insects, e.g. cockroaches and stick insects, have found fascinating solutions for the problem of locomotion, especially climbing over a large variety of obstacles. Research on behavioral neurobiology has identified key behavioral patterns of these animals (i.e., body flexion, center of mass elevation, and local leg reflexes) necessary for climbing. Inspired by this finding, we develop a neural control mechanism for hexapod robots which generates basic walking behavior and especially enables them to effectively perform reactive climbing behavior. The mechanism is composed of three main neural circuits: locomotion control, reactive backbone joint control, and local leg reflex control. It was developed and tested using a physical simulation environment, and was then successfully transferred to a physical six-legged walking machine, called AMOS II. Experimental results show that the controller allows the robot to overcome obstacles of various heights (e.g., ∼ 75% of its leg length, which are higher than those that other comparable legged robots have achieved so far). The generated climbing behavior is also comparable to the one observed in cockroaches.