Sound source localization (SSL) adds a spatial dimension to auditory perception, allowing a system to pinpoint the origin of speech, machinery noise, warning tones, or other acoustic events, capabilities that facilitate robot navigation, human–machine dialogue, and condition monitoring. While existing surveys provide valuable historical context, they typically address general audio applications and do not fully account for robotic constraints or the latest advancements in deep learning. This review addresses these gaps by offering a robotics-focused synthesis, emphasizing recent progress in deep learning methodologies. We start by reviewing classical methods such as time difference of arrival (TDOA), beamforming, steered-response power (SRP), and subspace analysis. Subsequently, we delve into modern machine learning (ML) and deep learning (DL) approaches, discussing traditional ML and neural networks (NNs), convolutional neural networks (CNNs), convolutional recurrent neural networks (CRNNs), and emerging attention-based architectures. The data and training strategy that are the two cornerstones of DL-based SSL are explored. Studies are further categorized by robot types and application domains to facilitate researchers in identifying relevant work for their specific contexts. Finally, we highlight the current challenges in SSL works in general, regarding environmental robustness, sound source multiplicity, and specific implementation constraints in robotics, as well as data and learning strategies in DL-based SSL. Also, we sketch promising directions to offer an actionable roadmap toward robust, adaptable, efficient, and explainable DL-based SSL for next-generation robots.