Recent neutron scattering experiments on the microscopic dynamics of polymers below and above the glass transition temperature T-g are reviewed. The results presented cover different dynamic processes appearing in glasses : local motions, vibrations, and different relaxation processes such as alpha- and beta-relaxation. For the alpha-relaxation, which occurs above T-g, it is possible to extend the time-temperature superposition principle, which is valid for polymers on a macroscopic scale, to the microscopic time scale. However, this principle is not applicable for temperatures approaching T-g. Below T-g, an inelastic excitation at a frequency of some hundred gigahertz (on the order of several wave numbers), the "boson peak," survives from a quasi-elastic overdamped scattering law at high temperatures. The connection between this boson peak and the fast dynamic process appearing near T-g is discussed.