I will describe the problem of designing robot motion planning algorithms for mechanical systems and explain its relation to the homotopy theory. I will survey the major properties of TC(X) and mention some interesting open problems, including problems of equivariant homotopy theory. I will state an interesting open conjecture about the recurrent nature of the homotopy invariants TC_r(X). It has been verified in the case of right angled Artin groups where a connection with the polyhedral products plays a role.

Part of the talk is based on a joint work with John Oprea.