  Manifolds occur naturally as configuration spaces of robotic systems. They
provide global descriptions of local coordinate systems that are common tools
in expressing positions of robots. The purpose of this survey is threefold.
Firstly, we present an overview over various results on topological
complexities of manifolds and related topics. Several constructions for
manifolds, e.g. symplectic structures and connected sums, can be used to
compute or estimate topological complexities. Secondly, we take a look at
geodesic motion planning in Riemannian manifolds. In this setting, results from
Riemannian geometry are employed to estimate the complexity of motion planning
along shortest paths in manifolds. Thirdly, we will discuss results on
connections between critical point theory and the topology of manifolds that
are related to motion planning problems. Here, we consider the role of
navigation functions for topological complexity and outline their relations to
newer numerical homotopy invariants, namely spherical complexities.
