The flow of a river in a channel of slowly varying width is investigated using an asymptotic approach. The work was motivated by a recent experimental investigation of this problem. The river transports sediment as bedload and is susceptible to an instability which causes variations in the depth of the river. The asymptotic theory is, in the first instance, used to describe steady-state flows in channels of varying width and it is found to give excellent agreement with experimental observations on this flow. The theory shows conclusively that a river of slowly increasing width will form central bars. Secondly, the approach is used to investigate the instability of the flow. The major result obtained is that in a symmetric channel which diverges from a width where the flow is unstable to one where instability is possible, then the preferred mode of instability is likely to be a central bar rather than an alternating bar as is the case in straight channels.