Residue-residue coevolution has been observed across a number of protein-protein interfaces, but the extent of residue coevolution between protein families on the whole-proteome scale has not been systematically studied. We investigate coevolution between 5.4 million pairs of proteins in Escherichia coli and between 3.9 millions pairs in Mycobacterium tuberculosis. We find strong coevolution for binary complexes involved in metabolism and weaker coevolution for larger complexes playing roles in genetic information processing. We take advantage of this coevolution, in combination with structure modeling, to predict protein-protein interactions (PPIs) with an accuracy that benchmark studies suggest is considerably higher than that of proteome-wide two-hybrid and mass spectrometry screens. We identify hundreds of previously uncharacterized PPIs in E. coli and M. tuberculosis that both add components to known protein complexes and networks and establish the existence of new ones.