The Weingarten calculus is a method for computing integrals of polynomial functions over compact classical Lie groups. The modern formulation was developed by Collins and Sniady in the 2000s, building on earlier work of the physicist Donald Weingarten. For unitary groups, the method translates polynomial integrals into finite sums over symmetric groups involving so-called Weingarten functions (analogous formulas exist for other compact classical groups). The basic mechanism behind the unitary case is the Schur–Weyl duality. I will explain the method and present some examples.