Statistical physics is a fertile source of high-dimensional partial differential equations. We shall survey recent developments concerning a system of nonlinear partial differential equations, which involves the Navier--Stokes system coupled with a high-dimensional parabolic Fokker--Planck equation describing the motion of polymer molecules in a viscous fluid occupying a bounded spatial domain. The model arises in the kinetic theory of dilute solutions of nonhomogeneous polymeric liquids, where the polymer molecules are idealized as bead-spring chains with finitely or infinitely extensible nonlinear elastic spring potentials, and has been the subject of active research over the past decade. We shall review recent results concerning the existence of large-data global weak solutions to this high-dimensional system. We shall also highlight a number of nontrivial open questions concerning the mathematical analysis, approximation and numerical analysis of high-dimensional Navier--Stokes--Fokker--Planck systems.