Recent advances in artificial intelligence have witnessed the emergence of large-scale deep learning models capable of interpreting and generating both textual and imaging data. Such models, typically referred to as foundation models (FMs), are trained on extensive corpora of unlabeled data and demonstrate high performance across various tasks. FMs have recently received extensive attention from academic, industry, and regulatory bodies. Given the potentially transformative impact that FMs can have on the field of radiology, radiologists must be aware of potential pathways to train these radiology-specific FMs, including understanding both the benefits and challenges. Thus, this review aims to explain the fundamental concepts and terms of FMs in radiology, with a specific focus on the requirements of training data, model training paradigms, model capabilities, and evaluation strategies. Overall, the goal of this review is to unify technical advances and clinical needs for safe and responsible training of FMs in radiology to ultimately benefit patients, providers, and radiologists.