The dynamics of atactic polystyrene (PS) has been studied by a quasielastic neutron scattering technique in a temperature range of 21-475 K covering the glass transition temperature T-g (=373 K). The so-called fast process in picosecond order is observed similarly to other glass-forming materials in supercooled states so far reported. However, the onset temperature of the fast process is similar to 170 K below T-g in PS, which is far below T-g in contrast to the previous observations. This result implies that the fast process is not necessarily the beta process predicted by the recently noted mode coupling theory. By comparing the result of hydrogenated PS with that of partially deuterated polystyrene (PS-d(5)) where only phenyl rings were preferentially deuterated, the origin of the fast process occurring far below T-g has been assigned to librational motion of phenyl rings, which is coupled with main chain motion near T-g. Analysis of the spectra by curve fit has suggested that the nature of the fast process changes at a temperature similar to 30 K above T-g. This change also involves the appearance of a slow relaxation process in the energy range of several tens mu eV. The origin of the slow process is discussed in terms of conformational transitions of the main chain.