Classification of protein structure has had a broad impact on our understanding of biological function and evolution, yet this work has largely focused on individual protein domains and their pairwise interactions. In contrast, the assembly of individual polypeptides into protein complexes, which are ubiquitous in cells, has received comparatively little attention. The periodic table of protein complexes is a new framework for analysis of complexes based on the principles of self-assembly. This reveals that sequence-identical subunits almost always have identical assembly roles within a complex and allows us to unify the vast majority of complexes of known structure (~32,000) into about 120 topologies. This facilitates the exhaustive enumeration of unobserved protein complex topologies and has significant practical applications for quaternary structure prediction, modelling and engineering.