Security-constrained unit commitment (SCUC) has been extensively studied as a key decision-making tool to determine optimal power generation schedules in the operation of electricity market. With the development of emerging power grids, fruitful research results on SCUC have been obtained. Therefore, it is essential to review current work and propose future directions for SCUC to meet the needs of developing power systems. In this paper, the basic mathematical model of the standard SCUC is summarized, and the characteristics and application scopes of common solution algorithms are presented. Customized models focusing on diverse mathematical properties are then categorized and the corresponding solving methodologies are discussed. Finally, research trends in the field are prospected based on a summary of the state-of-the-art and latest studies. It is hoped that this paper can be a useful reference to support theoretical research and practical applications of SCUC in the future.