Abstract—The accelerating advancements in Generative Artificial Intelligence (GenAI) have led to an unprecedented surge in data creation on the Internet, posing challenges to current computing and communication frameworks. GenAI, a distinct category of AI, generates content akin to human creations. Currently, GenAI services heavily rely on traditional cloud computing, resulting in high latency due to data transmission and a surge in requests. In response, the integration of edge-cloud computing emerges as an attractive paradigm, offering computation power and low latency through collaborative systems. This research paper provides a comprehensive overview of the intersection between GenAI and edge-cloud computing. We delve into recent developments in both domains and examine technical challenges through the lens of two exemplary GenAI applications. Introducing an innovative solution, we propose the Generative AI-oriented synthetical network (EcoGen), a collaborative cloud-edge-end intelligence framework. EcoGen facilitates bidirectional knowledge flow, allowing GenAI's pre-training to provide foundational knowledge for Edge Intelligence (EI), while EI aggregates personalized knowledge for GenAI. The framework leverages data-free knowledge relay to buffer contradictions, enabling virtuous-cycle model fine-tuning and task inference. Importantly, we incorporate a detailed analysis of the energy efficiency and environmental sustainability aspects of deploying Generative AI systems at scale, particularly in edge computing. Strategies to optimize energy consumption and reduce the carbon footprint are explored, contributing to a more sustainable AI ecosystem. Experimental results demonstrate the effectiveness of EcoGen in achieving seamless fusion and collaborative evolution between GenAI and EI. The paper concludes by outlining design considerations for training and deploying GenAI systems at scale and pointing towards future research directions, emphasizing the imperative of sustainable AI practices.