We prove the existence of global-in-time large-data weak solutions to a general class of models that arise from the kinetic theory of dilute solutions of nonhomogeneous polymeric liquids,where the polymer molecules are idealized as bead-spring chains with finitely extensible nonlinear elastic(FENE)type spring potentials.The class of models under consideration involves the unsteady,compressible,isentropic,isothermal Navier-Stokes system in a bounded domain in two or three space dimensions for the density $\rho$,the velocity $\ut$ and the pressure $p$ of the fluid,with an equation of state of the form $p(\rho)= c_p \rho^\gamma$,where $c_p$ is a positive constant and $\gamma>\frac{3}{2}$.The right-hand side of the Navier-Stokes momentum equation includes an elastic extra-stress tensor,which is the sum of the classical Kramers expression and a quadratic interaction term.The elastic extra-stress tensor stems from the random movement of the polymer chains and is defined through the associated probability density function that satisfies a Fokker-Planck-type parabolic equation,a crucial feature of which is the presence of a centre-of-mass diffusion term.This is joint work with John W.Barrett(Department of Mathematics,Imperial College London).