Positioning enhancement remains a pivotal technology in the evolution of both the fifth-generation advanced (5G-A) and 6G mobile communication networks, expected to offer users broader coverage, reduced latency, and more accurate and efficient positioning services based on 5G infrastructure. While in scenarios with inadequate coverage due to the sparse deployment of (both fixed and mobile) terrestrial base stations (BSs) or poor signal quality, uncrewed aerial vehicles (UAVs) can be flexibly deployed on a large scale. These UAVs act as edge reference BSs for 6G, providing on-demand positioning and other sensing-related services, such as logistics tracking and emergency search. In this article, we provide an extensive overview of UAV-assisted positioning architectures and summarize recent research advances, complete with detailed examples that illuminate the system operational processes. We also investigate key open issues and future research directions in UAV-assisted network systems, incorporating new 6 G features. This article outlines a road map for advancing various communication and sensing applications critical to 6G edge intelligence.