The rapid development of communication and computing techniques enables unmanned aerial vehicles (UAVs) to provide reliable and cost-effective wireless communication and computing services from the air. Compared to the conventional fixed infrastructure, UAVs have attractive attributes, such as high flexibility and operability, and, as a result, on-demand line-of-sight connection links. Therefore, UAV-assisted wireless networks have been envisioned as a promising paradigm to achieve enhanced coverage and connectivity for future wireless communications. Meanwhile, achieving high levels of energy efficiency, sensing, communication, and computing capacities, and security and privacy are critical to the success of UAV-assisted wireless networks. In order to improve the performance of UAV-assisted wireless networks, some frameworks and mechanisms have been developed in the past few years. In this article, we provide a comprehensive survey of these developments. Specifically, we conduct a brief overview for the architecture of UAV-assisted wireless networks from four domains (i.e., framework-related, technology-related, challenge-related, and solution-related) and four aspects (i.e., sensing-related, communication-related, computing-related, and application-related). Then, the integrated sensing, communication, and computing for UAV-assisted wireless networks is introduced, followed by the characteristics and requirements. We also provide the implementation and applications of UAV-assisted wireless networks. Next, we discuss the challenges and the state-of-the-art solutions for UAV-assisted wireless networks. Finally, the advanced technologies for UAV-assisted communication and computing networks are exploited, followed by the potential research directions.