This talk is an exposition of topological, and geometric properties of the classical configuration space of distinct particles on a manifold, as well as braid groups associated to the manifold.

The main setting is how features of these spaces 'connect' to several phenomena such as linking of circles in three dimensions, knots in three dimensions as well as homotopy groups of spheres. Explanations will be given for how and why these structures fit together.

One naive open question is whether the geometry of the configuration space of k particles in the two-sphere impacts related delicate combinatorial questions concerning certain specific topological invariants.