This paper introduces a family of error-correcting codes called zigzag codes. A zigzag code is described by a highly structured zigzag graph. Due to the structural properties of the graph, very low-complexity soft-in, soft-out decoding rules can be implemented. We present a decoding rule, based on the Max-Log-APP(MLA) formulation, which requires a total of only 20 addition-equivalent-operations per information bit per iteration. Simulation of a rate-1/2 concatenated zigzag code with four constituent encoders with interleaver length 65536 yields a bit error rate (BER) and of 10-5 at 0.9 dB and 1.4 dB away from the Shannon limit by optimal (APP) and low-cost sub-optimal (MLA) decoders, respectively.