Sonar systems are widely used for underwater sensing applications such as navigation, target detection, and environmental monitoring. However, conventional sonar signal processing techniques often rely on handcrafted features and model-based assumptions, which can limit performance in complex and noisy underwater environments. In recent years, learning-based approaches, particularly those using neural networks, have gained increasing attention as a data-driven alternative for sonar signal processing. These methods enable automatic feature extraction and improved adaptability across different operating conditions. This paper presents a review of learning-based sonar signal processing techniques, with emphasis on neural network models applied to single-receiver and multi-receiver sonar systems. Existing studies are categorized into feature-based deep learning approaches, end-to-end learning methods operating on raw or minimally processed acoustic data, and learning-based techniques for hydrophone array processing. The review also discusses general applications of deep learning in underwater acoustic sensing, including sonar imaging and target classification. Key advantages, limitations, and trends within each category are analyzed to provide a structured understanding of the current research landscape. Furthermore, this review highlights common challenges such as data scarcity, computational constraints, and limited generalization across environments. By organizing prior work and identifying research gaps, this paper aims to support the development of robust, data-driven sonar systems and to guide future research in learning-based underwater acoustic signal processing.