Medical images are naturally associated with rich semantics about the human anatomy, reflected in an abundance of recurring anatomical patterns, offering unique potential to foster deep semantic representation learning and yield semantically more powerful models for different medical applications. But how exactly such strong yet free semantics embedded in medical images can be harnessed for self-supervised learning remains largely unexplored. To this end, we train deep models to learn semantically enriched visual representation by self-discovery, self-classification, and self-restoration of the anatomy underneath medical images, resulting in a semantics-enriched, general-purpose, pre-trained 3D model, named Semantic Genesis. We examine our Semantic Genesis with all the publicly-available pre-trained models, by either self-supervision or fully supervision, on the six distinct target tasks, covering both classification and segmentation in various medical modalities (i.e., CT, MRI, and X-ray). Our extensive experiments demonstrate that Semantic Genesis significantly exceeds all of its 3D counterparts as well as the de facto ImageNet-based transfer learning in 2D. This performance is attributed to our novel self-supervised learning framework, encouraging deep models to learn compelling semantic representation from abundant anatomical patterns resulting from consistent anatomies embedded in medical images. Code and pre-trained Semantic Genesis are available at https://github.com/JLiangLab/SemanticGenesis.