As the next-generation wireless communication system, Sixth-Generation (6G) technologies are emerging, enabling various mobile edge networks that can revolutionize wireless communication and connectivity. By integrating Generative Artificial Intelligence (GAI) with mobile edge networks, generative mobile edge networks possess immense potential to enhance the intelligence and efficiency of wireless communication networks. In this article, we propose the concept of generative mobile edge networks and overview widely adopted GAI technologies and their applications in mobile edge networks. We then discuss the potential challenges faced by generative mobile edge networks in resource-constrained scenarios. To address these challenges, we develop a universal resource-efficient generative incentive mechanism framework, in which we design resource-efficient methods for network overhead reduction, formulate appropriate incentive mechanisms for the resource allocation problem, and utilize Generative Diffusion Models (GDMs) to find the optimal incentive mechanism solutions. Furthermore, we conduct a case study on resource-constrained mobile edge networks, employing model partition for efficient AI task offloading and proposing a GDM-based Stackelberg model to motivate edge devices to contribute computing resources for mobile edge intelligence. Finally, we propose several open directions that could contribute to the future popularity of generative mobile edge networks.