The development of applications based on artificial intelligence and implemented over wireless networks is increasingly rapidly and is expected to grow dramatically in the future. The resulting demand for the aggregation of large amounts of data has caused serious communication bottlenecks in wireless networks and particularly at the network edge. Over-the-air federated learning (OTA-FL), leveraging the superposition feature of multi-access channels (MACs), enables users at the network edge to share spectrum resources and achieves efficient and low-latency global model aggregation. This paper provides a holistic review of progress in OTA-FL and points to potential future research directions. Specifically, we classify OTA-FL from the perspective of system settings, including single-antenna OTA-FL, multi-antenna OTA-FL, and OTA-FL with the aid of the emerging reconfigurable intelligent surface (RIS) technology, and the contributions of existing works in these areas are summarized. Moreover, we discuss the trust, security and privacy aspects of OTA-FL, and highlight concerns arising from security and privacy. Finally, challenges and potential research directions are discussed to promote the future development of OTA-FL in terms of improving system performance, reliability, and trustworthiness. Specifical challenges to be addressed include model distortion under channel fading, the ineffective OTA aggregation of local models trained on substantially unbalanced data, and the limited accessibility and verifiability of individual local models.