The objective of this article is to provide a detailed review of the state-of-the-art underwater surveillance process and the new trends of the same. Underwater surveillance has recently getting lots of attention because of its potential applications including the security of the coastal border, effective fish farming, deep-sea exploration, preservation of rare aquatic animals, etc. In the case of underwater, the light that is sensed by the camera got degraded due to many underwater phenomena such as haze, low illumination, scattering, absorption, diffraction, and refraction. The actual color of the object, as well as the scene, gets degraded as the light can not travel deep underwater. In a situation, where the scene of view is not clear, it is very difficult to detect the moving objects present in the scene. Further, the identification of the object of concern and tracking of that becomes more challenging. In this survey, we categorize underwater surveillance as a combination of the three blocks: enhancement, object detection, and object tracking. We categorize all these three blocks based on the underwater complexity and motion models considered. In this article, we tried to enumerate the most detailed descriptions of each category. We also discuss the future directions of research in the area of underwater surveillance.