2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/59443We outline the recent classification of differential structures for all main classes of quantum groups. We also outline the algebraic notion of `quantum manifold' and `quantum Riemannian manifold' based on quantum group principal bundles, a formulation that works over general unital algebras.10 pages, to appear Proc. LMS Symposium Quantum Groups, Durham, 1999. This intro for mathematiciansQuantum Algebratext