Artificial Intelligence (AI) is transforming domains from healthcare and agriculture to finance and industry. As progress on Earth meets growing constraints, the next frontier is outer space, where AI can enable autonomous, resilient operations under extreme uncertainty and limited human oversight. This paper introduces Space AI as a unified interdisciplinary field at the intersection of artificial intelligence and space science and technology. We consolidate historical developments and contemporary progress, and propose a systematic framework that organises Space AI into four mission contexts: 1 AI on Earth, covering intelligent mission planning, spacecraft design optimisation, simulation, and ground-based data analytics; 2 AI in Orbit, focusing on satellite and station autonomy, space robotics, on-board/near-real-time data processing, communication optimisation, and orbital safety; 3 AI in Deep Space, enabling autonomous navigation, adaptive scientific discovery, resource mapping, and long-duration human-AI collaboration under communication constraints; and 4 AI for Multi-Planetary Life, supporting in-situ resource utilisation, habitat and infrastructure construction, life-support and ecological management, and resilient interplanetary networks. Ultimately, Space AI can accelerate humanity's capability to explore and operate in space, while translating advances in sensing, robotics, optimisation, and trustworthy AI into broad societal impact on Earth.