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 {\it 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.Comment: We have corrected three small typos on pages 8 and
