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Nanowire networks represent a unique class of neuromorphic systems. Their self-assembly confers a complex structure to their network circuitry, embedding a higher interconnectivity of resistive switching memory (memristive) cross-point junctions than can be achieved with top-down nanofabrication methods. Coupling of the nonlinear memristive dynamics to the network topology enables intrinsic adaptiveness and gives rise to emergent non-local dynamics. In this article, we summarise the physical principles underlying the memristive junctions and network dynamics of neuromorphic nanowire networks and provide the first comprehensive review of studies to date. We conclude with a perspective on future prospects for neuromorphic information processing.