Decomposition spaces, incidence algebras, and Moebius inversion I'll survey recent work with Imma Galvez and Andy Tonks [arXiv:1404.3202] developing an infinity-version of the theory of incidence algebras and Moebius inversion. The 'combinatorial objects' playing the role of posets and Moebius categories are DECOMPOSITION SPACES, simplicial $\infty$-groupoids satisfying an exactness condition weaker than the Segal condition, expressed in terms of generic and free maps in Delta. Just as the Segal condition expresses up-to-homotopy composition, the new condition expresses decomposition. The role of vector spaces is played by slices over infinity-groupoids, eventually with homotopy finiteness conditions imposed. To any decomposition space (subject to a completeness condition weaker than the Rezk condition) there is associated an incidence (co)algebra (with coefficients in infinity-groupoids), which satisfies an objective Moebius inversion principle in the style of Lawvere-Menni. Generic examples of decomposition spaces beyond Segal spaces are given by the Waldhausen S-construction (yielding Hall algebras) and by Schmitt restriction species, and many examples from classical combinatorics admit uniform descriptions in this framework. My focus will be on a specific example, namely the Lawvere-Menni Hopf algebra of Moebius intervals, which contains the universal Moebius function, but does not itself come from a Moebius category in the classical sense. It turns out that it DOES come from a decomposition space, which is in some sense universal. (The notion of decomposition space is equivalent to the notion of unital 2-Segal space of Dyckerhoff-Kapranov [arXiv:1212.3563], who develop other aspects of the theory, motivated by geometry, representation theory and homological algebra.)