Deep learning methods have revolutionised our ability to predict protein structures, allowing us a glimpse into the entire protein universe. As a result, our understanding of how protein structure drives function is now lagging behind our ability to determine and predict protein structure. Here, we describe how topology, the branch of mathematics concerned with qualitative properties of spatial structures, provides a lens through which we can identify fundamental organising features across the known protein universe. We identify topological determinants that capture global features of the protein universe, such as domain architecture and binding sites. Additionally, our analysis identifies highly specific properties, so-called topological generators, that can be used to provide deeper insights into protein structure-function and evolutionary relationships. We present a practical methodology for mapping the topology of the known protein universe at scale. We then use our approach to determine structural, functional and disease consequences of mutations. Our approach reveals and helps to explain differences in properties of proteins in mesophiles and thermophiles, and the likely structural and functional consequences of polymorphisms in a protein. For eukaryotes we find striking differences between protein topologies in multi-cellular and single-celled organisms.