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This paper presents the current state of mathematical modelling of the electrochemical behaviour of lithium-ion batteries as they are charged and discharged. It reviews the models developed by Newman and co-workers, both in the cases of dilute and moderately-concentrated electrolytes and indicates the modelling assumptions required for their development. Particular attention is paid to the interface conditions imposed between the electrolyte and the active electrode material; necessary conditions are derived for one of these, the Butler-Volmer relation, in order to ensure physically realistic solutions. Insight into the origin of the differences between various models found in the literature is revealed by considering formulations obtained by using different measures of the electric potential. Materials commonly used for electrodes in lithium ion batteries are considered and the various mathematical models used to describe lithium transport in them discussed. The problem of up-scaling from models of behaviour at the single electrode particle scale to the cell scale is addressed using homogenisation techniques resulting in the pseudo 2D model commonly used to describe charge transport and discharge behaviour in lithium-ion cells. Numerical solution to this model is discussed and illustrative results for a common device are computed.