Despite technological and medical advances, the detection, interpretation, and treatment of cancer based on imaging data continue to pose significant challenges. These include high inter-observer variability, difficulty of small-sized lesion detection, nodule interpretation and malignancy determination, inter-and intra-tumour heterogeneity, class imbalance, segmentation inaccuracies, and treatment effect uncertainty. The recent advancements in Generative Adversarial Networks (GANs) in computer vision as well as in medical imaging may provide a basis for enhanced capabilities in cancer detection and analysis. In this review, we assess the potential of GANs to address a number of key challenges of cancer imaging, including data scarcity and imbalance, domain and dataset shifts, data access and privacy, data annotation and quantification, as well as cancer detection, tumour profiling and treatment planning. We provide a critical appraisal of the existing literature of GANs applied to cancer imagery, together with suggestions on future research directions to address these challenges. We analyse and discuss 163 papers that apply adversarial training techniques in the context of cancer imaging and elaborate their methodologies, advantages and limitations. With this work, we strive to bridge the gap between the needs of the clinical cancer imaging community and the current and prospective research on GANs in the artificial intelligence community.