Abstract: I'll start briefly with the classical theory of incidence algebras for posets (Rota 1963) and Leroux's generalisation to certain categories called Mobius categories (1975). A key element in this theory is Mobius inversion, a counting device exploiting how combinatorial objects can be decomposed. Then I will survey recent work with Imma Galvez and Andy Tonks developing a far-reaching generalisation to something called decomposition spaces (or 2-Segal spaces [Dyckerhoff-Kapranov]). There are three directions of generalisation involved: firstly, the theory is made objective, meaning that it works with the combinatorial objects themselves, rather than with vector spaces spanned by them. This can be seen as a systematic way of turning algebraic proofs into bijective proofs. The role of vector spaces is played by slice categories. Secondly, the theory incorporates homotopy theory by passing from categories to infinity-categories in the form of Segal spaces. (This is relevant even for classical combinatorics to deal with symmetries.) Finally, the Segal condition is replaced by something weaker (decomposition spaces): where the Segal condition expresses composition, the new condition expresses decomposition. This allows to cover a wide range of combinatorial Hopf algebras that cannot directly be the incidence algebra of any poset or Mobius category, such as the Butcher-Connes-Kreimer Hopf algebra of trees, or Schmitt's chromatic Hopf algebra of graphs. It also turns out to have interesting connections to representation theory, covering all kinds of Hall algebras: the Waldhausen S-construction of an abelian category is an example of a decomposition space. I will finish with the general Mobius inversion principle for decomposition spaces. Throughout I will stress the general ideas behind, avoiding technicalities.