Semantic communication leverages artificial intelligence (AI) technologies to extract semantic information from data for efficient transmission, theraby significantly reducing communication cost. With the evolution towards artificial general intelligence (AGI), the increasing demands for AGI services pose new challenges to semantic communication. In response, we propose a new paradigm for AGI-driven communications, called generative semantic communication (GSC), which utilizes advanced AI technologies such as foundation models and generative models. We first describe the basic concept of GSC and its difference from existing semantic communications, and then introduce a general framework of GSC, followed by two case studies to verify the advantages of GSC in AGI-driven applications. Finally, open challenges and new research directions are discussed to stimulate this line of research and pave the way for practical applications.