Despite technological and medical advances, the detection, interpretation, and treatment of cancer based on imaging data continue to pose significant challenges. These include inter-observer variability, class imbalance, dataset shifts, inter- and intra-tumour heterogeneity, malignancy determination, and treatment effect uncertainty. Given the recent advancements in image synthesis, Generative Adversarial Networks (GANs), and adversarial training, we assess the potential of these technologies to address a number of key challenges of cancer imaging. We categorise these challenges into (a) data scarcity and imbalance, (b) data access and privacy, (c) data annotation and segmentation, (d) cancer detection and diagnosis, and (e) tumour profiling, treatment planning and monitoring. Based on our analysis of 164 publications that apply adversarial training techniques in the context of cancer imaging, we highlight multiple underexplored solutions with research potential. We further contribute the Synthesis Study Trustworthiness Test (SynTRUST), a meta-analysis framework for assessing the validation rigour of medical image synthesis studies. SynTRUST is based on 26 concrete measures of thoroughness, reproducibility, usefulness, scalability, and tenability. Based on SynTRUST, we analyse 16 of the most promising cancer imaging challenge solutions and observe a high validation rigour in general, but also several desirable improvements. 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 data synthesis and adversarial networks in the artificial intelligence community.