Recent advances in federated learning (FL) have brought large-scale collaborative machine learning opportunities for massively distributed clients with performance and data privacy guarantees. However, most current works focus on the interest of the central controller in FL and overlook the interests of the FL clients. This may result in unfair treatment of clients, which discourages them from actively participating in the learning process and damages the sustainability of the FL ecosystem. Therefore, the topic of ensuring fairness in FL is attracting a great deal of research interest. In recent years, diverse fairness-aware FL (FAFL) approaches have been proposed in an effort to achieve fairness in FL from different perspectives. However, there is no comprehensive survey that helps readers gain insight into this interdisciplinary field. This article aims to provide such a survey. By examining the fundamental and simplifying assumptions, as well as the notions of fairness adopted by the existing literature in this field, we propose a taxonomy of FAFL approaches covering major steps in FL, including client selection, optimization, contribution evaluation, and incentive distribution. In addition, we discuss the main metrics for experimentally evaluating the performance of FAFL approaches and suggest promising future research directions toward FAFL.