Artificial Intelligence brings exciting innovations in all aspects of life and creates new opportunities across industry sectors. At the same time, it raises significant questions in terms of trust, ethics, and accountability. This paper offers an introduction to the AI4Media project, which aims to build on recent advances of AI in order to offer innovative tools to the media sector. AI4Media unifies the fragmented landscape of media-related AI technologies by investigating new learning paradigms and distributed AI, exploring issues of AI explainability, robustness and privacy, examining AI techniques for content analysis, and exploiting AI to address major societal challenges. In this paper, we focus on our vision of how such AI technologies can reshape the media sector, by discussing seven industrial use cases that range from combating disinformation in social media and supporting journalists for news story creation, to high quality video production, game design, and artistic co-creation. For each of these use cases, we highlight the present challenges and needs, and explain how they can be efficiently addressed by using innovative AI-driven solutions.