The recent advancements in artificial intelligence (AI) combined with the extensive amount of data generated by today's clinical systems, has led to the development of imaging AI solutions across the whole value chain of medical imaging, including image reconstruction, medical image segmentation, image-based diagnosis and treatment planning. Notwithstanding the successes and future potential of AI in medical imaging, many stakeholders are concerned of the potential risks and ethical implications of imaging AI solutions, which are perceived as complex, opaque, and difficult to comprehend, utilise, and trust in critical clinical applications. Despite these concerns and risks, there are currently no concrete guidelines and best practices for guiding future AI developments in medical imaging towards increased trust, safety and adoption. To bridge this gap, this paper introduces a careful selection of guiding principles drawn from the accumulated experiences, consensus, and best practices from five large European projects on AI in Health Imaging. These guiding principles are named FUTURE-AI and its building blocks consist of (i) Fairness, (ii) Universality, (iii) Traceability, (iv) Usability, (v) Robustness and (vi) Explainability. In a step-by-step approach, these guidelines are further translated into a framework of concrete recommendations for specifying, developing, evaluating, and deploying technically, clinically and ethically trustworthy AI solutions into clinical practice.