In recent years, with the great success of compressed sensing (CS) in the field of signal processing, matrix completion (MC), derived from CS, has increasingly become a hot research topic in the field of machine learning. Many researchers have done a lot of fruitful studies on matrix completion problem modeling and their optimization, and constructed relatively complete matrix completion theory. In order to better grasp the development process of matrix completion, and facilitate the combination of matrix completion theory and engineering applications, this article reviews the existing matrix completion models and their algorithms. First, it introduces the natural evolution process from CS to MC, and illustrates that the development of CS theory has laid the foundation for the formation of MC theory. Second, the article summarizes the existing matrix completion models into the four classes from the perspective of the relaxation of non-convex and non-smooth rank function, aiming to provide reasonable solutions for specific matrix completion applications; Third, in order to understand the inherent optimization techniques and facilitate solving new problem-dependent matrix completion model, the article studies the representative optimization algorithms suitable for various matrix completion models. Finally, article analyzes the existing problems in current matrix completion technology, proposes possible solutions for these problems, and discusses the future work.