General approaches for designing sequence-specific peptide binding proteins would have wide utility in proteomics and synthetic biology. Although considerable progress has been made in designing proteins which bind to other proteins, the general peptide binding problem is more challenging as most peptides do not have defined structures in isolation, and to offset the loss in solvation upon binding the protein binding interface has to provide specific hydrogen bonds that complement the majority of the buried peptide backbone polar groups. Inspired by natural repeat protein-peptide complexes, and engineering efforts to alter their specificity, we describe a general approach for de novo design of proteins made out of repeating units that bind peptides with repeating sequences such that there is a one to one correspondence between repeat units on the protein and peptide. We develop a rapid docking plus geometric hashing method to identify protein backbones and protein-peptide rigid body arrangements that are compatible with bidentate hydrogen bonds between side chains on the protein and the backbone of the peptide; the remainder of the protein sequence is then designed using Rosetta to incorporate additional interactions with the peptide and drive folding to the desired structure. We use this approach to design, from scratch, alpha helical repeat proteins that bind six different tripeptide repeat sequences--PLP, LRP, PEW, IYP, PRM and PKW-- in near polyproline 2 helical conformations. The proteins are expressed at high levels in E. coli, are hyperstable, and bind peptides with 4-6 copies of the target tripeptide sequences with nanomolar to picomolar affinities both in vitro and in living cells. Crystal structures reveal repeating interactions between protein and peptide interactions as designed, including a ladder of protein side chain to peptide backbone hydrogen bonds. By redesigning the binding interfaces of individual repeat units, specificity can be achieved for non-repeating sequences, and for naturally occurring proteins containing disordered regions. Our approach provides a general route to designing specific binding proteins for a broad range of repeating and non-repetitive peptide sequences.