Microformal or thick morphisms, introduced by Ted Voronov, are a generalisation of smooth maps between manifolds that still give rise to pullbacks on functions. These pullbacks are in general nonlinear and formal, and in special cases they define L-infinity morphisms between the algebras of functions on homotopy Poisson or homotopy Schouten manifolds. In this talk, I will give a brief introduction to thick morphisms (of which there are so called classical and quantum versions), and describe a graphical calculus which calculates all terms in the formal power series that result from their pullbacks. The method heavily resembles the use of Feynman diagrams in perturbative quantum field theory, and the relationship between the calculi for classical and quantum thick morphisms is exactly the relationship between the tree-level and full perturbative expansions in QFT.