This paper comprehensively reviews recent advances in underwater acoustic signal denoising, an area critical for improving the reliability and clarity of underwater communication and monitoring systems. Despite significant progress in the field, the complex nature of underwater environments poses unique challenges that complicate the denoising process. We begin by outlining the fundamental challenges associated with underwater acoustic signal processing, including signal attenuation, noise variability, and the impact of environmental factors. The review then systematically categorizes and discusses various denoising algorithms, such as conventional, decomposition-based, and learning-based techniques, highlighting their applications, advantages, and limitations. Evaluation metrics and experimental datasets are also reviewed. The paper concludes with a list of open questions and recommendations for future research directions, emphasizing the need for developing more robust denoising techniques that can adapt to the dynamic underwater acoustic environment.