Polymer-processing operations with dominating elongational flow have a great relevance, especially in several relevant industrial applications. Film blowing, fiber spinning and foaming are some examples in which the polymer melt is subjected to elongational flow during processing. To gain a thorough knowledge of the material-processing behavior, the evaluation of the rheological properties of the polymers experiencing this kind of flow is fundamental. This paper reviews the main achievements regarding the processing-structure-properties relationships of polymer-based materials processed through different operations with dominating elongational flow. In particular, after a brief discussion on the theoretical features associated with the elongational flow and the differences with other flow regimes, the attention is focused on the rheological properties in elongation of the most industrially relevant polymers. Finally, the evolution of the morphology of homogeneous polymers, as well as of multiphase polymer-based systems, such as blends and micro- and nano-composites, subjected to the elongational flow is discussed, highlighting the potential and the unique characteristics of the processing operations based on elongation flow, as compared to their shear-dominated counterparts.