In polymer solutions or blends, flow can strongly influence the degree of mixing of the components. In a shearing flow, droplets in a dispersion can be broken down to sizes comparable to the dimensions of the polymer molecules themselves, thereby inducing molecular-scale mixing. Demixing can also occur when the two components of the mixture differ greatly in viscoelastic properties. Shear or extensional flow can induce polymer migration in nonhomogeneous flows or in flows with curved streamlines, and can render turbid solutions or blends that are otherwise transparent. Flow can also induce polymer gelation, and can induce ordering transitions in liquid crystals or block copolymers. Here, we review these phenomena, discuss proposed mechanisms, and assess the degree to which recent theories can account for the observations. Because the phenomena are complex, multiple experimental probes and theoretical methods are required to study them. Successful theories must incorporate polymer/polymer or polymer/solvent thermodynamics, critical phenomena, and phase transitions, as well as polymer rheology and the kinetics of diffusion or crystallization. The experimental techniques used to study these phenomena are equally wide ranging, and include turbidity measurements, light, x-ray, and neutron scattering, fluorescence quenching, microscopy, and rheology.