Decompositions spaces, incidence algebras, and Moebius inversion I'll explain how the classical Leroux theory of incidence algebras of Moebius categories (which covers both locally finite posets (Rota) and finite-decomposition monoids (Cartier-Foata)) admits a far-reaching generalisation in terms of what we call decomposition spaces (introduced independently by Dyckerhoff-Kapranov). They are simplicial (infinity) groupoids satisfying an exactness condition weaker than the Segal condition. Just as the Segal condition expresses up-to-homotopy composition, the new condition expresses decomposition. Specific new examples covered by the theory include the Faa di Bruno and Connes-Kreimer bialgebras, and the Lawvere-Menni Hopf algebra of Moebius intervals which contains the universal Moebius function (but does not itself come from a Moebius category). Generic classes of examples include Schmitt incidence algebras of restriction species, and Hall algebras: the Waldhausen S-construction of abelian (or stable infinity) categories are decomposition spaces, and their incidence algebras are Hall algebras. While explaining the basic theory and the key examples mentioned above, I will also spend some time explaining the bigger programme of upgrading some aspects of enumerative and algebraic combinatorics from finite sets to homotopy-finite groupoids and (infinity-groupoids), and how recent progress in higher category theory allows for this upgrade at a very reasonable price. This is joint work with Imma Galvez and Andy Tonks.