The Federal Aviation Administration (FAA) recently introduced a new standard, namely, remote identification, to improve accountability for unmanned aerial vehicles (UAVs) operations. This rule requires UAV operators to broadcast messages revealing sensitive data, such as identity and location on the wireless channel. However, this leads to security and privacy concerns among UAV operators. Unauthorized parties may easily discover the location and identity of a UAV flying in a specific area and launch attacks on it such as using wireless jamming or tracking its activity. This review investigates and systematizes the main weaknesses affecting the Remote ID capability required of modern UAVs, and the approaches through which attackers can exploit these weaknesses to disrupt safety and accountability. Moreover, this article analyzes current solutions that mitigate privacy issues associated with Remote ID. Finally, we identify multiple challenges that require to be addressed by both industry and academia, and we propose future research directions to improve the security and privacy of UAVs.