Artificial Intelligence (AI) promises solutions to the challenges raised by the digitalization of power grids and their assets. Decision-making, forecasting and even operational optimization of grids and assets are just some of the solutions that AI algorithms can provide to operators, utilities and vendors. Nevertheless, barriers such as access to quality datasets, interpretability, repeatability, and availability of computational resources currently limit the extent of practical AI implementations. At the same time, Digital Twins (DTs) are foreseen as platforms that can overcome these barriers, and also provide a new environment for the development of enhanced and more intelligent applications. In this manuscript, we review the published literature to determine the existing capabilities and implementation challenges of AI algorithms in power systems, and classify AI-based applications based on their time scale to reveal their temporal sensitivity. Furthermore, DT-based technologies are discussed, identifying the potentials to tackle current limitations of real-world AI applications as well as exploring the synergies between DTs and AI. By combining AI and DT, we outline multiple prospective use cases for AI-enhanced power grid and power asset DTs. Our review also identifies that the combination of AI-based solutions and DTs leverages new applications with the potential to fundamentally change multiple aspects of the power industry.