Deep learning (DL) is currently the standard artificial intelligence tool for computer-based image analysis in radiology. Traditionally, DL models have been trained with strongly supervised learning methods. These methods depend on reference standard labels, typically applied manually by experts. In contrast, weakly supervised learning is more scalable. Weak supervision comprises situations in which only a portion of the data are labeled (incomplete supervision), labels refer to a whole region or case as opposed to a precisely delineated image region (inexact supervision), or labels contain errors (inaccurate supervision). In many applications, weak labels are sufficient to train useful models. Thus, weakly supervised learning can unlock a large amount of otherwise unusable data for training DL models. One example of this is using large language models to automatically extract weak labels from free-text radiology reports. Here, we outline the key concepts in weakly supervised learning and provide an overview of applications in radiologic image analysis. With more fundamental and clinical translational work, weakly supervised learning could facilitate the uptake of DL in radiology and research workflows by enabling large-scale image analysis and advancing the development of new DL-based biomarkers.