The development of positioning systems has been significantly advanced by a combination of technological innovations, such as improved sensors, signal processing, and computational power, alongside inspiration drawn from biological mechanisms. Although vision is the main means for positioning oneself—or elements relative to oneself—in the environment, other sensory mediums provide additional information, and may even take over when visibility is lacking, such as in the dark or in troubled waters. In particular, the auditory system in mammals greatly contributes to determining the location of sound sources, as well as navigating or identifying objects’ texture and shape, when combined with echolocation behavior. Taking further inspiration from the neuronal processing in the brain, neuromorphic computing has been studied in the context of sound source localization and echolocation-based navigation, which aim at better understanding biological processes or reaching state-of-the-art performances in energy efficiency through the use of spike encoding. This paper sets out a review of these neuromorphic sound source localization, sonar- and radar-based navigation systems, from their earliest appearance to the latest published works. Current trends and possible future directions within this scope are discussed.