The details of the molecular mechanisms underlying subglass relaxations in amorphous polymers remain largely a mystery. Although there is a considerable body of useful experimental description, the experiments are rarely specific in revealing the nature of the motions taking place. Molecular modelling or simulation is becoming of increasing value in interpreting the details of molecular motion. An important recent advance has the been the ability to generate, via molecular dynamics (MD) simulation, detailed description of the structure and packing in amorphous polymeric melts and glasses. In the present work, such advances in modelling will be exploited in describing subglass relaxations. The work will be done in association with dielectric relaxation experiments. Systems to be studied include subglass processes in main-chain liquid crystalline aromatic polyesters; subglass processes in other aromatic polyesters; the gramma- relaxation in polyethylene; subglass processes that are, by design, at very low temperature by virtue of low internal rotational barriers; and, side-chain mesogen liquid crystalline polymers. Unlike many other materials, such as metals and ceramics, the usefulness of all polymeric materials depends on the fact that they are very dynamic on the molecular scale. That is, the constituent molecules are in a constant state of motion and the kinds of motion determine the properties. Even in polymers that are apparently rigid and glass-like, residual molecular motions determine such behavior as strength and toughness. However there is not yet a clear understanding of these motions. The present work is directed toward attaining this understanding in order that better materials can be developed. Modern computer modelling techniques will be combined with sophisticated experimental tools in accomplishing this goal.