Recent advancements in unmanned aerial vehicles (UAVs) have proven UAVs to be an inevitable part of future networking and communications systems. While many researchers have proposed UAV-assisted solutions for improving traditional network performance by extending coverage and capacity, an in-depth study on aspects of artificial intelligence-based autonomous UAV network design has not been fully explored yet. The objective of this paper is to present a comprehensive survey of AI-based autonomous UAV networks. A careful survey was conducted of more than 100 articles on UAVs focusing on the classification of autonomous features, network resource management and planning, multiple access and routing protocols, and power control and energy efficiency for UAV networks. By reviewing and analyzing the UAV networking literature, it is found that AI-based UAVs are a technologically feasible and economically viable paradigm for cost-effectiveness in the design and deployment of such next-generation autonomous networks. Finally, this paper identifies open research problems in the emerging field of UAV networks. This study is expected to stimulate more research endeavors to build low-cost, energy-efficient, next-generation autonomous UAV networks.