A computer model for meandering rivers that couples water flow, bed topography, the sorting of sediments with different grain sizes, and channel migration has been developed using the linear theory of Johannesson and Parker  for the dynamics of meandering rivers and the theory of Parker and Andrews  for bed load sediment transport and sorting in meander bends. The equations for single-size sediment transport used in the linear theory of Johannesson and Parker were replaced by multiple-size sediment transport and sorting equations obtained from the theory of Parker and Andrews. This paper is the first of a pair. In this paper, the theoretical framework of the model is presented, and the derivation of the theory from the earlier theories is detailed. It is shown that there are five different regions in the two-dimensional model parameter space [ϵ(G1 − 3), (π/2)2 G2], where G1 is a measure of the coupling of the longitudinal sediment flow to the longitudinal water flow, ϵ is the rescaled bed friction coefficient, and G2 characterizes the coupling of the transverse (cross channel) sediment flow to the transverse component of the bed inclination. The model is not stable in two of the regions (regions 1 and 2). Alternating bars develop in region 4 but propagate along the channel with damped amplitudes, and no alternating bars develop in region 5. In region 3, which is the boundary between regions 2 and 4, alternating bars develop and propagate undamped throughout the entire channel. Increasing the flöodplain inclination, channel width/depth ratio, average sediment size, or the breadth of the sediment size distribution in the model increases the tendency toward a transition from a stable meandering state to an unstable, presumably braided, state.