With the rapid development of 3D scanning and processing technologies, point cloud surface reconstruction has become a significant research focus in the fields of computer graphics and computer vision. The goal of point cloud surface reconstruction is to recover a continuous surface of an object or scene from discrete and irregular point cloud. In recent years, implicit function has gained increasing attention for their robustness and flexibility. Notably, the introduction of deep learning has significantly improved their performance in reconstructing complex geometries. A growing body of research on the application of implicit functions for point cloud surface reconstruction has emerged, making it increasingly challenging to track and understand the broader landscape. Therefore, there is an urgent need for a comprehensive review to summarize the fundamental principles, key techniques, and developmental trajectories of these methods. This paper first reviews the fundamental principles and historical development of implicit functions in point cloud surface reconstruction. It then systematically surveys existing methods from six perspectives: the definition of implicit functions, types of implicit functions, technological advancements, loss functions, datasets and evaluation metrics. Finally, the paper identifies critical challenges in this field, such as handling low-quality point clouds, ensuring real-time performance and addressing sequential point cloud reconstruction. It also provides an outlook on future research directions. This review aims to serve as a comprehensive technical reference for researchers engaged in point cloud reconstruction and related fields, helping them gain an in-depth understanding of the field's dynamics and efficiently identify frontier problems.