Emerging spatial computing systems seamlessly superimpose digital information on the physical environment observed by a user, enabling transformative experiences across various domains, such as entertainment, education, communication and training^{, –}. However, the widespread adoption of augmented-reality (AR) displays has been limited due to the bulky projection optics of their light engines and their inability to accurately portray three-dimensional (3D) depth cues for virtual content, among other factors^,. Here we introduce a holographic AR system that overcomes these challenges using a unique combination of inverse-designed full-colour metasurface gratings, a compact dispersion-compensating waveguide geometry and artificial-intelligence-driven holography algorithms. These elements are co-designed to eliminate the need for bulky collimation optics between the spatial light modulator and the waveguide and to present vibrant, full-colour, 3D AR content in a compact device form factor. To deliver unprecedented visual quality with our prototype, we develop an innovative image formation model that combines a physically accurate waveguide model with learned components that are automatically calibrated using camera feedback. Our unique co-design of a nanophotonic metasurface waveguide and artificial-intelligence-driven holographic algorithms represents a significant advancement in creating visually compelling 3D AR experiences in a compact wearable device.