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Curvilinear object segmentation is a paramount step for many applications ranging from medical to aerial image processing. In particular, vessel segmentation in retinal images, detection of spiculated lesions in mammograms or extraction of airways in CT scans provide essential information to experts to evaluate, diagnose and propose a treatment. The significance of these applications has conducted important efforts to propose curvilinear object segmentation algorithms based on the most different techniques. The main objective of this review is to clearly present the similarities and differences between curvilinear structures in different applications and the different techniques used to segment them more effectively. To do so, we propose a general definition of curvilinear structures that encompasses the distinct models considered in the literature. In addition, we analyse and classify the mathematical techniques …