There is growing excitement about the potential of leveraging artificial intelligence (AI) to tackle some of the outstanding barriers to the full deployment of robots in daily lives. However, action and sensing in the physical world pose greater and different challenges for AI than analysing data in isolation and it is important to reflect on which AI approaches are most likely to be successfully applied to robots. Questions to address, among others, are how AI models can be adapted to specific robot designs, tasks and environments. This Perspective offers an assessment of what AI has achieved for robotics since the 1990s and proposes a research roadmap with challenges and promises. These range from keeping up-to-date large datasets, representatives of a diversity of tasks that robots may have to perform, and of environments they may encounter, to designing AI algorithms tailored specifically to robotics problems but generic enough to apply to a wide range of applications and transfer easily to a variety of robotic platforms. For robots to collaborate effectively with humans, they must predict human behaviour without relying on bias-based profiling. Explainability and transparency in AI-driven robot control are essential for building trust, preventing misuse and attributing responsibility in accidents. We close with describing what are, in our view, primary long-term challenges, namely, designing robots capable of lifelong learning, and guaranteeing safe deployment and usage, as well as sustainable development.