Federated learning (FL) becomes popular and has shown great potentials in training large-scale machine learning (ML) models without exposing the owners’ raw data. In FL, the data owners can train ML models based on their local data and only send the model updates rather than raw data to the model owner for aggregation. To improve learning performance in terms of model accuracy and training completion time, it is essential to recruit sufficient participants. Meanwhile, the data owners are rational and may be unwilling to participate in the collaborative learning process due to the resource consumption. To address the issues, there have been various works recently proposed to motivate the data owners to contribute their resources. In this paper, we provide a comprehensive review for the economic and game theoretic approaches proposed in the literature to design various schemes for incentivizing data owners to participate in FL training process. In particular, we first present the fundamentals and background of FL, economic theories commonly used in incentive mechanism design. Then, we review applications of game theory and economic approaches applied for incentive mechanisms design of FL. Finally, we highlight some open issues and future research directions concerning incentive mechanism design of FL.