In this study, 83 proteins containing helix–loop–helix–loop repeats were designed—with sequences unrelated to known repeat proteins—and experimentally characterized; 43 solution X-ray scattering spectra and 15 structures of the designed proteins show that these non-natural repeat proteins have a broad range of curvatures and that their overall structures are in close agreement with design models. Repeat proteins are composed of multiple tandem copies of a modular structure unit and are widespread in nature, playing critical roles in molecular recognition, signalling, and other essential biological processes. In natural repeat proteins, the interactions between adjacent units define the shape and curvature of the overall structure. Two papers published in this issue of Nature describe the design of geometrically unconstrained, open tandem repeat arrays. David Baker and colleagues used computational protein design to generate a series of proteins containing repeats of a simple 'helix-loop-helix-loop' structural motif. Data from 43 proteins with solution X-ray scattering spectra, and 15 structures of the designed proteins, show that these non-natural repeat proteins have a broad range of curvatures and that their overall structures are in close agreement with design models. Philip Bradley and colleagues used computational protein design to synthesize a series of alpha-solenoid/toroid structures that have various radii and different sized 'holes'. The authors solved X-ray crystal structures of four of the designed proteins and determined that their overall structures are in close agreement with the design models. A central question in protein evolution is the extent to which naturally occurring proteins sample the space of folded structures accessible to the polypeptide chain. Repeat proteins composed of multiple tandem copies of a modular structure unit1 are widespread in nature and have critical roles in molecular recognition, signalling, and other essential biological processes2. Naturally occurring repeat proteins have been re-engineered for molecular recognition and modular scaffolding applications3,4,5. Here we use computational protein design to investigate the space of folded structures that can be generated by tandem repeating a simple helix–loop–helix–loop structural motif. Eighty-three designs with sequences unrelated to known repeat proteins were experimentally characterized. Of these, 53 are monomeric and stable at 95 °C, and 43 have solution X-ray scattering spectra consistent with the design models. Crystal structures of 15 designs spanning a broad range of curvatures are in close agreement with the design models with root mean square deviations ranging from 0.7 to 2.5 Å. Our results show that existing repeat proteins occupy only a small fraction of the possible repeat protein sequence and structure space and that it is possible to design novel repeat proteins with precisely specified geometries, opening up a wide array of new possibilities for biomolecular engineering.