The comparison of the three-dimensional shapes of protein molecules poses a complex algorithmic problem. Its solution provides biologists with computational tools to organize the rapidly growing set of thousands of known protein shapes, to identify new types of protein architecture, and to discover unexpected evolutionary relations, reaching back billions of years, between protein molecules. Protein shape comparison also improves tools for identifying gene functions in genome databases by defining the essential sequence-structure features of a protein family. Finally, an exhaustive all-on-all shape comparison provides a map of physical attractor regions in the abstract shape space of proteins, with implications for the processes of protein folding and evolution.