As a specific category of artificial intelligence (AI), generative artificial intelligence (GenAI) generates new content that resembles what is created by humans. The rapid development of GenAI systems has created a huge amount of new data on the Internet, posing new challenges to current computing and communication frameworks. Currently, GenAI services rely on the traditional cloud computing framework due to the need for large computation resources. However, such services will encounter high latency because of data transmission and a high volume of requests. On the other hand, edge-cloud computing can provide adequate computation power and low latency at the same time through the collaboration between edges and the cloud. Thus, it is attractive to build GenAI systems at scale by leveraging the edge-cloud computing paradigm. In this overview paper, we review recent developments in GenAI and edge-cloud computing, respectively. Then, we use two exemplary GenAI applications to discuss technical challenges in scaling up their solutions using edge-cloud collaborative systems. Finally, we list design considerations for training and deploying GenAI systems at scale and point out future research directions.