Smart grid is the new trend for clean, sustainable, efficient and reliable energy generation, delivery and use. To ensure stable and secure operation is essential for the smart grid, which needs effective stability analysis and control. As the smart grid has evolved through a growing scale of interconnection, increasing integration of renewable energy, widespread operation of direct current power transmission systems, and liberalization of electricity markets, the stability characteristics of it are much more complex than the past. Due to these changes, conventional stability analysis and control approaches have a series of drawbacks in terms of speed, effectiveness and economy. On the contrary, the emerging artificial intelligence (AI) techniques provide powerful and promising tools for stability analysis and control in smart grids and have attracted growing attention. This paper aims to give a comprehensive and clear picture of recent advances in this research area. First, we present a general overview of AI, including its definitions, history and state-of-the-art methodologies. And then, this paper gives a comprehensive review of its applications to security assessment, stability assessment, fault diagnosis, and stability control in smart grids. These applications have achieved impressive results. Nevertheless, we also identify some major challenges these applications face in practice: high requirements on data, imbalanced learning, interpretability of AI, difficulties in transfer learning, the robustness of AI to communication quality, and the robustness against attack or adversarial examples. Furthermore, we provide suggestions for potential important future investigation directions to overcome these challenges and bridge the gap between research and practice. •An overview of state-of-the-art artificial intelligence (AI) is provided.•A review of AI’s applications to smart grid stability and control is undertaken.•This paper identifies some major challenges these applications face in practice.•Suggestions and potentially important future research directions are put forward.