Mathematical modelling fluvial flow, sediment transport and morphological evolution started half a century ago and, to date, a variety of mathematical models have been developed and are in widespread use. However, the quality of mathematical river modelling remains uncertain because of: (a) poor assumptions in model formulations; (b) simplified numerical solution procedure; (c) the implementation of sediment relationships of questionable validity; and (d) the problematic use of model calibration and verification as assertions of model veracity. An overview of mathematical models for alluvial rivers is provided in this and the companion paper 'Part 2: Special issues'. This paper is the first part, providing a general review of mathematical river models. The issues addressed comprise what have been obvious since the very beginning of mathematical river modelling and are still open, and also the pertinent components that pose challenges to model developers and end-users pursuing refined modelling practice. In particular the simplified mass conservation equations, asynchronous solution procedures, sediment transport functions, movable-bed resistance, numerical difficulty for strong hyperbolic equations, and representation of movable and complex geometry are discussed. A test example is provided to demonstrate the impacts of simplified mass conservation equations and an asynchronous solution procedure in comparison with those of largely tuned friction factors. It is concluded that mathematical models for fluvial flow-sediment-morphology systems are far from being mature, and that considerable expertise, physical insight and experience are vital for meaningful solutions to be acquired and for the limitations of modelling outputs to be properly identified, interpreted and assessed.