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Recently, unmanned aerial vehicles (UAVs) attracted significant popularity in both military and civilian domains for various applications and services. Moreover, UAV-aided wireless sensor networks (UAWSNs) became one of the interesting hot research topics. This is mainly because UAWSNs can significantly increase the network coverage and energy utilization compared to traditional wireless sensor networks (WSNs). However, the high mobility, dynamic path, and variable altitude of UAVs can cause not only unforeseen changes in the network topology but also connectivity and coverage problems, which can affect the routing performance of the network. Therefore, the design of a routing protocol for UAWSNs is a critical task. In this paper, the routing protocols for UAWSNs are extensively investigated and discussed. Firstly, we classify the existing routing protocols based on different network criteria. They are extensively reviewed and compared with each other in terms of advantages and limitation, routing metrics and policies, characteristics, difference performance factors, and different performance optimization factors. Furthermore, open research issues and challenges are summarized and discussed.