2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/215314We 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.We have corrected three small typos on pages 8 and 9Geometric TopologyDifferential Geometry57N15; 57N75; 57P99; 53C70text