Vortices are commonly understood as rotating motions in fluid flows. The analysis of vortices plays an important role in numerous scientific applications, such as in engineering, meteorology, oceanology, medicine and many more. The successful analysis consists of three steps: vortex definition, extraction and visualization. All three have a long history, and the early themes and topics from the 70s survived to this day, namely the identification of vortex cores, their extent and the choice of suitable reference frames. This paper provides an overview over the advances that have been made in the last forty years. We provide sufficient background on differential vector field calculus, extraction techniques like critical point search and the parallel vectors operator, and we introduce the notion of reference frame invariance. We explain the most important region-based and line-based methods, integration-based and geometry-based approaches, recent objective techniques, the selection of reference frames by means of flow decompositions, as well as a recent local optimization-based technique. We point out relationships between the various approaches, classify the literature and identify open problems and challenges for future work.
