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Optical communications is undergoing a digital revolution, as digital signal processing (DSP) emerges as a practical solution for robust long-haul transmission. In contrast to previous systems, in which optical dispersion compensation was considered a necessity, recently uncompensated long-haul transmission has been demonstrated using DSP at the transmitter, receiver, or a combination of the two. We review the historic developments of electronic signal processing as applied to optical communications before focusing the latest developments, namely digital coherent systems. We then discuss the salient features of the electronic signal processing schemes currently under investigation before discussing the challenges and opportunities offered b...