The benefits of tracking, identifying, measuring features of interest from structure responses have endless applications for saving cost, time and improving safety. To date, structural health monitoring (SHM) has been extensively applied in several fields, such as aerospace, automotive, and mechanical engineering. However, the focus of this paper is to provide a comprehensive up-to-date review of civil engineering structures such as buildings, bridges, and other infrastructures. For this reason, this article commences with a concise introduction to the fundamental definitions of SHM. The next section presents the general concepts and factors that determine the best strategy to be employed for SHM. Afterward, a thorough review of the most prevalent anomaly detection approaches, from classic techniques to advanced methods, is presented. Subsequently, some popular benchmarks, including laboratory specimens and real structures for validating the proposed methodologies, are demonstrated and discussed. Finally, the advantages and disadvantages of each method are summarized, which can be helpful in future studies.