In recent years, flying ad hoc networks (FANET), formed from unmanned aerial vehicles (UAVs), have absorbed the attention of academic and industrial research communities due to their many applications in military and civilian fields. FANETs benefit from unique features, including highly moving UAVs and dynamic topological structure. Therefore, most existing routing protocols, such as the greedy perimeter stateless routing (GPSR), are not compatible with the FANET environment and its specific features. To improve the performance of GPSR in FANET, it is important to address several challenges, namely the selection of the right period for broadcasting hello messages in the network, the selection of the right criteria for selecting the next-hop node, and the improvement of reliability in the data transfer process. In this paper, an adaptive and multi-path greedy perimeter stateless routing (AM-GPSR) protocol is suggested in FANETs. It includes two new strategies, namely adaptive hello strategy and multi-path greedy forwarding strategy. The adaptive hello strategy defines a special hello broadcast period for each UAV according to its speed and error between two estimated and actual positions. Furthermore, the greedy forwarding strategy carries out a filtering operation on candidate nodes and eliminates border UAVs and those that are far from the destination. Then, candidate UAVs are prioritized based on the time to reach the destination and buffer capacity, and UAVs with higher priorities are chosen to send data packets. Finally, AM-GPSR applies a greedy multi-path forwarding strategy to increase reliability in the data transmission process. Lastly, the simulation of AM-GPSR is done via the network simulator version 2 (NS2) to evaluate its performance. This evaluation process includes two different scenarios, i.e. change in the speed of UAVs and change in their communication range. In this process, AM-GPSR is compared with three other methods, namely the aerial greedy geographic routing (AGGR) protocol, the geolocation assisted aeronautical routing protocol (AeroRP), and GPSR. This comparison shows the successful performance of AM-GPSR in terms of delivery success rate, throughput, and delay. Although the control overhead of the proposed method is more than that of AGGR.