Machine learning algorithms are widely used in management systems in different fields, such as employee recruitment, loan provision, disease diagnosis, etc., and even in some risky decision-making areas, playing an increasingly crucial role in decisions affecting people’s lives and social development. However, the use of algorithms for automated decision-making can cause unintentional biases that lead to discrimination against certain specific groups. In this context, it is crucial to develop machine learning algorithms that are not only accurate but also fair. There is an extensive discussion of algorithmic fairness in the existing literature. Many scholars have proposed and tested definitions of fairness and attempted to address the problem of unfairness or discrimination in algorithms. This review aims to outline different definitions of algorithmic fairness and to introduce the procedure for constructing fair algorithms to enhance fairness in machine learning. First, this review divides the definitions of algorithmic fairness into two categories, namely, awareness-based fairness and rationality-based fairness, and discusses existing representative algorithmic fairness concepts and notions based on the two categories. Then, metrics for unfairness/discrimination identification are summarized and different unfairness/discrimination removal approaches are discussed to facilitate a better understanding of how algorithmic fairness can be implemented in different scenarios. Challenges and future research directions in the field of algorithmic fairness are finally concluded.