Human readers or radiologists routinely perform full-body multi-organ multi-disease detection and diagnosis in clinical practice, while most medical AI systems are built to focus on single organs with a narrow list of a few diseases. This might severely limit AI's clinical adoption. A certain number of AI models need to be assembled non-trivially to match the diagnostic process of a human reading a CT scan. In this paper, we construct a Unified Tumor Transformer (CancerUniT) model to jointly detect tumor existence & location and diagnose tumor characteristics for eight major cancers in CT scans. CancerUniT is a query-based Mask Transformer model with the output of multi-tumor prediction. We decouple the object queries into organ queries, tumor detection queries and tumor diagnosis queries, and further establish hierarchical relationships among the three groups. This clinically-inspired architecture effectively assists inter-and intra-organ representation learning of tumors and facilitates the resolution of these complex, anatomically related multi-organ cancer image reading tasks. CancerUniT is trained end-to-end using curated large-scale CT images of 10,042 patients including eight major types of cancers and occurring non-cancer tumors (all are pathology-confirmed with 3D tumor masks annotated by radiologists). On the test set of 631 patients, CancerUniT has demonstrated strong performance under a set of clinically relevant evaluation metrics, substantially outperforming both multi-disease methods and an assembly of eight single-organ expert models in tumor detection, segmentation, and diagnosis. This moves one step closer towards a universal high performance cancer screening tool.