Neutron spin echo spectroscopy (NSE) provides the unique opportunity to unravel the molecular dynamics of polymer chains in space and time, covering most of the relevant length and time scales. This article reviews in a comprehensive form recent advances in the application of NSE to problems in polymer physics and describes in terms of examples expected future trends. The review commences with a description of NSE covering both the generic longitudinal field set-up as well as the resonance technique. Then, NSE results for homopolymers chains are presented, covering all length scales from the very local secondary β-relaxation to large scale reptation. This overview is the core of the review. Thereafter the dynamics of more complex systems is addressed. Starting from polymer blends, diblock copolymers, gels, micelles, stars and dendrimers, rubbery electrolytes and biological macromolecules are discussed. Wherever possible the review relates the NSE findings to the results of other techniques, in particular emphasizing computer simulations.