The recent explosion of ‘big data’ has ushered in a new era of artificial intelligence (AI) algorithms in every sphere of technological activity, including medicine, and in particular radiology. However, the recent success of AI in certain flagship applications has, to some extent, masked decades-long advances in computational technology development for medical image analysis. In this article, we provide an overview of the history of AI methods for radiological image analysis in order to provide a context for the latest developments. We review the functioning, strengths and limitations of more classical methods as well as of the more recent deep learning techniques. We discuss the unique characteristics of medical data and medical science that set medicine apart from other technological domains in order to highlight not only the potential of AI in radiology but also the very real and often overlooked constraints that may limit the applicability of certain AI methods. Finally, we provide a comprehensive perspective on the potential impact of AI on radiology and on how to evaluate it not only from a technical point of view but also from a clinical one, so that patients can ultimately benefit from it.