The motion planning problem of robotics leads to an interesting invariant of topological spaces, topological complexity TC(X), depending on the homotopy type of the configuration space of the system. TC(X) is an integer reflecting the complexity of motion planning algorithms for all systems (robots) having X as their configuration space. Methods of algebraic topology allow computation of TC(X) in many examples of practical interest. In the case when the space X is aspherical the number TC(X) depends only on the fundamental group of X. I will describe some recent results expressing the topological complexity of aspherical spaces in the case when the fundamental group is hyperbolic in the sense of Gromov.

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