Unmanned aerial vehicles (UAVs) have attracted massive attention in many engineering and practical applications in the last years for their characteristics and operation flexibility. For the UAV system, suitable control systems are required to operate appropriately and efficiently. An emerging control technique is visual servoing utilizing the onboard camera systems for inspecting the UAV's environment and autonomously controlling the UAV's operation. Artificial intelligence (AI) techniques are widely deployed in the visual servoing of autonomous UAV applications. Despite the increasing research in the field of AI-based visual control of UAV systems, comprehensive review articles that showcase the general trends and future directions in this field of research are limited. This work comprehensively examines the application and advancements of AI-enhanced visual servoing in autonomous UAV systems, covering critical control tasks and offering insights into future research directions for enhancing performance and applicability which is limited in the current literature. The paper first reviews the application of intelligent visual servoing systems for autonomously executing various UAV control tasks, including 3D UAV positioning, aerial and ground object following, obstacle avoidance, and autonomous landing. Second, the research progresses in applying AI techniques in the visual servoing of autonomous UAV systems are discussed and analyzed. Finally, future directions and critical research gaps for further improving the performance and applicability of intelligent visual servoing systems are included.