A classical result of von Neumann states that if f is an analytic function on the open unit disc in the complex plane which is bounded in absolute value by 1, and if T is a bounded linear operator on a Hilbert space which is a contraction (the norm of T is less then or equal to 1), then f(T) is a contraction. (If you are worried about substituting an operator into an analytic function, you may assume f is a polynomial; you may even assume T is a finite matrix.) This result lies at the root of several key concepts in operator theory and operator algebras, such as dilations, operator models, and complete contractivity. It is also intimately related to integral representations and interpolation problems for analytic functions of one complex variable, as well as to scattering theory (the Lax--Phillips scheme) and to system theory (conservative systems and their transfer functions). After reviewing these classical connections I will describe some more recent results generalizing some of these ideas to the setting of several complex variables and beyond. The lecture is aimed at a general mathematical audience and should be accessible for graduate students.