After defining polynomial functors and introducing their calculus in terms of certain bridge diagrams, I will survey some examples related to logic: Girard's normal functors, Jay and Cockett's shapely types, the containers of Abbott, Altenkirch and Ghani, multicategories after Lambek, Burroni, and Leinster, and finally trees. (This talk is based on a paper in preparation with Nicola Gambino.)