Title: Mšbius inversion in general categories Abstract: I will report on recent joint work with Imma G‡lvez and Andy Tonks on Mšbius inversion in general categories (with coefficients in groupoids). I will start by briefly recalling the classical theory, from Mšbius to Rota, and then explain the notion of Mšbius category of Leroux; here subtle finiteness conditions are required in order to get well-defined sums. The theory of Mšbius inversion for general categories is achieved by working with groupoids as coefficients instead of numbers. At the same time this turns all constructions and proofs "bijective" instead of algebraic. More precisely, the equations obtained are equivalences of groupoids, and the techniques have a homotopy flavour. Extracting algebraic identities is a matter to taking homotopy cardinality, and for this some finiteness is obviously required. It is a homotopy finiteness weaker than Leroux's, and covers examples that are not Mšbius categories in the sense of Leroux, for example categories of trees. The Connes-Kreimer Hopf algebra of trees arises directly as the incidence algebra of the category of trees, without the need of constructing auxiliary posets as done by DŸr.