The rheological properties and processing of polymer nanocomposites with dispersed nanometer-thick layered silicates are reviewed. The linear viscoelastic properties reflect the nanoscale dispersion of the silicate layers and the mesoscale structure that such dispersed layers or their aggregates form. The linear viscoelasticity exhibits time-temperature super-positioning, with the exception of systems in which dispersions are stabilized by hydrogen bonds. The non-linear stress relaxation behavior obeys time-strain separability, whereas large-amplitude oscillatory shear leads to strong alignment of the layers in the flow direction. The empirical Cox-Merz rule equating the steady shear viscosity and complex viscosity fails for these nanocomposites. It is conjectured that this results from orientation of the layers or their aggregates in response to a steady flow. Such orientation is also responsible for the silicate concentration independence of the first normal stress difference when compared at similar shear stress values.