General design principles for protein interaction specificity are challenging to extract. In DNA, specificity arises from a limited set of hydrogen-bonding interactions in the core of the double helix to design and build a wide range of shapes. In proteins, specificity arises largely from buried hydrophobic packing complemented by irregular peripheral polar interactions. Protein-based materials have the potential for even greater geometric and chemical diversity, including additional functionality. Here we describe a general approach for designing a wide range of protein oligomers that have interaction specificity determined by modular arrays of extensive hydrogen bond networks. We use the approach to design dimers, trimers, and tetramers consisting of two concentric rings of helices, including previously not seen triangular, square, and supercoiled topologies. X-ray crystallography confirms that the structures overall, and the hydrogen-bond networks in particular, are nearly identical to the design models, and the networks confer interaction specificity in vivo. The ability to design extensive hydrogen-bond networks with atomic accuracy enables the programming of protein interaction specificity for a broad range of synthetic biology applications; more generally, our results demonstrate that, even with the tremendous diversity observed in nature, there are fundamentally new modes of interaction to be discovered in proteins.