With the fast and unstoppable evolution of robotics and artificial intelligence, effective autonomous navigation in real-world scenarios has become one of the most pressing challenges in the literature. However, demanding requirements, such as real-time operation, energy and computational efficiency, robustness, and reliability, make most current solutions unsuitable for real-world challenges. Thus, researchers are forced to seek innovative approaches, such as bio-inspired solutions. Indeed, animals have the intrinsic ability to efficiently perceive, understand, and navigate their unstructured surroundings. To do so, they exploit self-motion cues, proprioception, and visual flow in a cognitive process to map their environment and locate themselves within it. Computational neuroscientists aim to answer ''how'' and ''why'' such cognitive processes occur in the brain, to design novel neuromorphic sensors and methods that imitate biological processing. This survey aims to comprehensively review the application of brain-inspired strategies to autonomous navigation, considering: neuromorphic perception and asynchronous event processing, energy-efficient and adaptive learning, or the imitation of the working principles of brain areas that play a crucial role in navigation such as the hippocampus or the entorhinal cortex.