A contractive (scalar, matrix or operator valued) analytic function on the unit disc in the complex plane can be represented in an essentially unique way as the transfer function of a conservative input/state/output linear system or as the scattering function of a scattering system (in the sense of Lax-Phillips and Adamyan-Arov). This representation is closely related to several central topics in operator theory and function theory: characteristic functions of contraction operators, unitary dilations and the von Neumann inequality, as well as classical interpolation problems. Following the seminal work of Agler in the late 1980s and the early 1990s (as well as independent work of Kummert at around the same time), there were considerable developments in generalizing these ideas to the setting of analytic functions on the polydisc in $C^d$. I will survey some of the concepts and results; a lot of the talk will be based on joint work with Joe Ball, Dmitry Kaliuzhnyi-Verbovetskyi, and our late colleague Cora Sadosky.