The purpose of this paper is to provide a structural review of the progress made on detection and localization of leaks in pipelines by using approaches based on the Kalman filter. This is, to the best of our knowledge, the first review on the t opic. In particular, it is the first to try to draw the attention of the leak detection community to the important contributions that use the Kalman filter as the core of a computational pipeline monitoring system. Without being exhaustive, we try to gather the results from different research groups and present them in a unified fashion. For this reason, we propose a classification of the current approaches based on the Kalman filter. For each of the existing approaches within this classification, the basic concepts, fundamental results, and relations with the other approaches are discussed in detail. The review starts from a short summary of basic concepts about state observers. Then, a brief history of the use of the Kalman filter for diagnosing leaks is described by mentioning the most outstanding approaches. At last, we briefly discuss some emerging research problems, such as the leak detection in pipelines transporting heavy oils, and we discuss the main challenges and some open problems.