Three lectures on polynomial functors In the first lecture, I'll develop some elementary aspects of the theory of polynomial functors. The main result will be Tambara's theorem, stating that the category of finite polynomial functors is the Lawvere theory for commutative semirings. I will also explain one interpretation of polynomial functors in type theory: their initial algebras are inductive data types. In the second lecture, the main themes will be trees and polynomial monads. While specific kinds of trees (P-trees) can be viewed as forming the initial algebra of a polynomial endofunctor 1+P, it is also possible to define trees more combinatorially: a tree is itself a certain polynomial endofunctor. This leads to the interpretation of P-trees as operations of the free monad on P, emphasising another aspect of trees, namely substitution. Polynomial monads are essentially operads, and the polynomial viewpoint turns out to be useful in handling the combinatorics underlying algebraic structures, for example to construct model structures on categories of algebras. Finally in the third lecture, I'll talk about combinatorial species and polynomial functors over groupoids and infinity groupoids, covering first the necessary background (which is not so much). Groupoid coefficients are needed to transparently handle combinatorial structures with symmetries, and in particular combinatorial structures with so many symmetries that they don't form classical species (such as Feynman graphs). Infinity-groupoid coefficients serve to capture higher homotopical data, such as inherent in intensional type theory.