Decomposition spaces, incidence algebras and Mšbius inversion I'll start rather leisurely with a review of incidence algebras and Mšbius inversion, starting with the classical Mšbius function in number theory, then incidence algebras and Mšbius inversion for locally finite posets (Rota) and monoids with the finite decomposition property (Cartier-Foata), and finally their common generalisation to Mšbius categories (Leroux). From here I'll move on to survey recent work with G‡lvez and Tonks taking these constructions into homotopy theory. On one hand we generalise from categories to Rezk-complete Segal spaces, taking an objective approach working directly with coefficients in infinity-groupoids instead of numbers. (Under certain finiteness conditions, numerical results can be obtained by taking homotopy cardinality.) On the other hand we show that the Segal condition is not needed for these constructions: it can be replaced by a weaker exactness condition formulated in terms of generic and free maps in Delta. This new notion we call decomposition space: while the Segal condition expresses up-to-homotopy composition, the new condition expresses decomposition. (An equivalent notion, formulated in terms of triangulations of polygons, was discovered independently by Dyckerhoff-Kapranov under the name unital 2-Segal space.) Many convolution algebras in combinatorics arise as the incidence algebra of a decomposition space which is not a category, for example those constructed by Schmitt from restriction species. The Waldhausen S-construction of an abelian (or stable infinity-) category is another example of a decomposition space; the associated incidence algebras are versions of (derived) Hall algebras. To finish I'll explain how the Lawvere-Menni Hopf algebra, which contains the universal Mšbius function, arises from a decomposition space of Mšbius intervals. [Reference: arXiv:1404.3202.]