The emerging area of Intelligent Digital Twins (IDTs) offers great potential as a new paradigm for accelerating scientific discovery, while also offering state-of-the-art functionality in controlling complex physical processes. We investigate this concept for the case of an Intelligent Digital Twin of metal additive manufacturing (AM). Metal AM is an excellent choice for utilising of an IDT due to the process being an inherently complex multi-physics one, with key elements including granular powder flow, laser melting and material solidification. This complexity means that computational simulations are extremely costly and obtaining high quality experimental data extremely difficult, so optimal exploration of the parameter space using all available information on the current uncertainty in the region of interest is highly desirable. Our Intelligent Digital Twin for this process includes a complete description of the target geometry of the object being printed and a set of data-driven and computational models for the different physical processes occurring in the system. The data-driven models consist of a set of Gaussian Processes (GP) that can be trained using combinations of real world sensor data and outputs from computational simulations. We illustrate the utility of our IDT by determining optimal input print parameters and obtaining Pareto fronts between competing priorities such as surface roughness and print time. We also demonstrate the potential of the IDT as an intelligent control system to respond to errors during the print process and dynamically improve final print quality.