The detection of echolocation clicks is key in understanding the intricate behaviors of cetaceans and monitoring their populations. Cetacean species relying on clicks for navigation, foraging and even communications are sperm whales (Physeter macrocephalus) and a variety of dolphin groups. Echolocation clicks are wideband signals of short duration that are often emitted in sequences of varying inter-click-intervals. While datasets and models for clicks exist, the detection and classification of clicks present a significant challenge, mostly due to the diversity of clicks' structures, overlapping signals from simultaneously emitting animals, and the abundance of noise transients from, for example, snapping shrimps and shipping cavitation noise. This paper provides a survey of the many detection and classification methodologies of clicks, ranging from 2002 to 2023. We divide the surveyed techniques into categories by their methodology. Specifically, feature analysis (e.g., phase, ICI and duration), frequency content, energy based detection, supervised and unsupervised machine learning, template matching and adaptive detection approaches. Also surveyed are open access platforms for click detections, and databases openly available for testing. Details of the method applied for each paper are given along with advantages and limitations, and for each category we analyze the remaining challenges. The paper also includes a performance comparison for several schemes over a shared database. Finally, we provide tables summarizing the existing detection schemes in terms of challenges address, methods, detection and classification tools applied, features used and applications.