Derivation of a mathematical model for microbial fuel cells (MFC) with suspended biomass and added electron-transfer mediator is described. The model is based on mass balances for several dissolved chemical species such as substrate, oxidized mediator and reduced mediator. Biological, chemical and electrochemical reactions can occur in the bulk liquid and at the electrode surface, respectively. Model outputs include time-dependent production of current and electrical charge, current-voltage and current-power curves, and the evolution of concentrations of chemical species. The model behaviour is illustrated using a test case based on detailed experimental observations reported in the literature for a microbial fuel cell operated in batch mode and repeatedly fed with a single substrate. A detailed model parameter estimation procedure is presented. The model simulates the current-time evolution and voltage-current curves in the MFC with glucose as anode substrate and the ferrocyanide/ferricyanide redox couple as the oxidation reaction at the cathode. Simulations show the effect of different parameters (electrical resistance, mass transfer resistance, exchange current, coulombic yields and biomass, substrate and mediator concentrations) on the MFC characteristics. The model explains how the endogenous metabolism or intracellular substrate storage could lead to a non-zero background current even when the added substrate has been depleted. Different trends (increasing or decreasing) in the initial current are explained by the initial oxidation state of the mediator (oxidized or reduced, respectively). The model has potential applications for other bioelectrochemical systems.