The use of unmanned aerial vehicles (UAVs) is attracting considerable interest in academic research, commercial, and military applications. Multi-UAV systems organized in an ad hoc fashion called a flying ad hoc network (FANET) can cooperatively and collaboratively accomplish complex missions more efficiently compared to single large UAV systems. However, the unique features of FANETs such as high mobility, low node density, and high frequency of topology changes introduce challenges to the communication design, especially routing. Thus, the routing requirements of FANETs surpass those of MANETs or VANETs. In this paper, we present UAV classification, communication and application architectures, and an exhaustive survey of the existing routing protocols for flying ad hoc networks. Furthermore, we highlight the key features, strengths and weaknesses, and different mobility models used for the performance evaluation of the existing FANET routing protocols. More importantly, a proposed taxonomy and a review on the existing FANET routing protocols are presented. Finally, we highlight the existing challenges and open research issues.