Artificial intelligence (AI), in one form or another, has been a part of medical imaging for decades. The recent evolution of AI into approaches such as deep learning has dramatically accelerated the application of AI across a wide range of radiologic settings. Despite the promises of AI, developers and users of AI technology must be fully aware of its potential biases and pitfalls, and this knowledge must be incorporated throughout the AI system development pipeline that involves training, validation, and testing. Grand challenges offer an opportunity to advance the development of AI methods for targeted applications and provide a mechanism for both directing and facilitating the development of AI systems. In the process, a grand challenge centralizes (with the challenge organizers) the burden of providing a valid benchmark test set to assess performance and generalizability of participants' models and the collection and curation of image metadata, clinical/demographic information, and the required reference standard. The most relevant grand challenges are those designed to maximize the open-science nature of the competition, with code and trained models deposited for future public access. The ultimate goal of AI grand challenges is to foster the translation of AI systems from competition to research benefit and patient care. Rather than reference the many medical imaging grand challenges that have been organized by groups such as MICCAI, RSNA, AAPM, and grand-challenge.org, this review assesses the role of grand challenges in promoting AI technologies for research advancement and for eventual clinical implementation, including their promises and limitations.