Linked from
The 24 pages that link to Configuration space, each with the reason it gives.
Rigid bodyNarrower topic: A rigid body's configurations form a space of positions and orientations.
Affine spaceRelated: Positions of unconstrained points often form affine configuration spaces.
Smooth manifoldRelated: Constraints often make a system's configuration space a smooth manifold.
Lonely runner conjectureRelated: All runner positions together form a moving point on a torus.