Healthcare data are inherently multimodal. Almost all data generated and acquired during a patient's life can be hypothesized to contain information relevant to providing optimal personalized healthcare. Data sources such as ECGs, doctor's notes, histopathological and radiological images all contribute to inform a physician's treatment decision. However, most machine learning methods in healthcare focus on single-modality data. This becomes particularly apparent within the field of radiology, which, due to its information density, accessibility, and computational interpretability, constitutes a central pillar in the healthcare data landscape and traditionally has been one of the key target areas of medically-focused machine learning. Computer-assisted diagnostic systems of the future should be capable of simultaneously processing multimodal data, thereby mimicking physicians, who also consider a multitude of resources when treating patients. Before this background, this review offers a comprehensive assessment of multimodal machine learning methods that combine data from radiology and other medical disciplines. It establishes a modality-based taxonomy, discusses common architectures and design principles, evaluation approaches, challenges, and future directions. This work will enable researchers and clinicians to understand the topography of the domain, describe the state-of-the-art, and detect research gaps for future research in multimodal medical machine learning.