The Bing-borsuk and the Busemann Conjectures

We present two classical conjectures concerning the characterization of manifolds: the Bing Borsuk Conjecture asserts that every n-dimensional homogeneous ANR is a topological n-manifold, whereas the Busemann Conjecture asserts that every n-dimensional G-space is a topological n-manifold. The key object in both cases are so-called generalized manifolds, i.e. ENR homology manifolds. We look at the history, from the early beginnings to the present day. We also list several open problems and related conjectures.