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Scour induced by currents is one of the main causes of the bridge failure in rivers. Fundamental knowledge and mechanisms on scour processes due to currents are often taken as a basis for scour studies, which are the focus of this review. Scour development induced by waves and in combined wave–current conditions are also briefly discussed. For the design of structure foundations, the maximum scour depths need to be estimated. The mechanisms of local scour and predictions of maximum local scour depths have been studied extensively for many years. Despite the complexity of the scour process, a lot of satisfying results and progresses have been achieved by many investigators. In order to get a comprehensive review of local scour for vertical piles, major progresses made by researchers are summarized in this review. In particular, maximum scour depth influencing factors including flow intensity, sediments, pile parameters and time are analyzed with experimental data. A few empirical equations referring to temporary scour depth and maximum scour depth were classified with their expressing forms. Finally, conclusions and future research directions are addressed.