In this thesis we study the homotopy invariant TC(X); the topological complexity of a space X. This invariant, introduced by Farber in [15], was originally motivated by a problem in Robotics; the motion planning problem. We study relations between the topological complexity of a space and its fundamental group, namely when the fundamental group is ”small”, i.e. either has small order or small cohomological dimension. We also apply the navigation functions technique introduced in [20] to the study of the topological complexity of projective and lens spaces. In particular, we introduce a class of navigation functions on projective and lens spaces. It is known ([25]) that the topological complexity of a real projective space equals one plus its immersion dimension. A similar approach to the immersion dimension of some lens spaces has been suggested in [31]. Finally, we study the topological complexity (and other invariants) of random right-angled Artin groups, i.e. the stochastic behaviour of the topological complexity of Eilenberg-MacLane spaces of type K(G, 1), where G is a right-angled Artin group associated to a random graph.