"Polynomial functors over groupoids, and combinatorial Dyson-Schwinger equations" Polynomial functors are essentially functors defined in terms of sums, products, and exponentiation. They can be seen as a categorification of elementary arithmetic, but they also constitute a general machinery for encoding and handling combinatorial structures and data types. Groupoid coefficients rather than set coefficients are needed to transparently handle symmetries, like those occurring in Feynman graphs. The real Dyson-Schwinger equation are the 'quantum equations of motion'. I will only talk about their combinatorial skeleton, Kreimer's combinatorial Dyson-Schwinger equations, which are fixpoint equations whose solutions are certain series of graphs or trees. I will explain how any polynomial endofunctor P generates a free monad, whose operations form a groupoid of P-trees, which is a solution to an abstract combinatorial Dyson-Schwinger equation X = 1+P(X), and satisfies a Faˆ di Bruno formula in the Connes-Kreimer bialgebra of P-trees. Analogous results for Feynman graphs are obtained by specialising to certain endofunctors P given in terms of interaction labels and 1PI primitive graphs. Many of the ideas involved originate in the theory of inductive data types. If time permits, I may say something about that.