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The success of major current and forthcoming rover missions now calls for significant efforts to mitigate this risk. Here, we review known processes that could have generated false biosignatures on early Mars. These examples are known largely from serendipitous discoveries rather than systematic research and remain poorly understood; they probably represent only a small subset of relevant phenomena. These phenomena tend to be driven by kinetic processes far from thermodynamic equilibrium, often in the presence of liquid water and organic matter, conditions similar to those that can actually give rise to, and support, life. We propose that strategies for assessing candidate biosignatures on Mars could be improved by new knowledge on the physics and chemistry of abiotic self-organization in geological systems. 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