The static and dynamic properties of polymer-layered silicate nanocomposites are discussed, in the context of polymers in confined spaces and polymer brushes. A wide range of experimental techniques as applied to these systems are reviewed, and the salient results from these are compared with a mean field thermodynamic model and non-equilibrium molecular dynamics simulations. Despite the topological constraints imposed by the host lattice, mass transport of the polymer, when entering the galleries defined by adjacent silicate layers, is quite rapid and the polymer chains exhibit mobilities similar to or faster than polymer self-diffusion. However, both the local and global dynamics of the polymer in these nanoscopically confined galleries are dramatically different from those in the bulk. On a local scale, intercalated polymers exhibit simultaneously a fast and a slow mode of relaxation for a wide range of temperatures, with a marked suppression (or even absence) of cooperative dynamics typically associated with the glass transition. On a global scale, relaxation of polymer chains either tethered to or in close proximity (<1nm as in intercalated hybrids) to the host surface are also dramatically altered. In the case of the tethered polymer nanocomposites, similarities are drawn to the dynamics of other intrinsically anisotropic fluids such as ordered block copolymers and smectic liquid crystals. Further, new non-linear viscoelastic phenomena associated with melt-brushes are reported and provide complementary information to those obtained for solution-brushes studied using the Surface Forces Apparatus.