Abstract A systematic understanding of traffic dynamics on road networks is crucial for many transportation studies and can help to develop more efficient ramp metering, evacuation, signal control, and other management and control strategies. In this study, we present a theory of multi-commodity network traffic flow based on the Lighthill–Whitham–Richards (LWR) model. In particular, we attempt to analyze kinematic waves of the Riemann problem for a general junction with multiple upstream and downstream links. In this theory, kinematic waves on a link can be determined by its initial condition and prevailing stationary state. In addition to a stationary state, a flimsy interior state can develop next to the junction on a link. In order to pick out unique, physical solutions, we introduce two types of entropy conditions in supply-demand space such that (i) speeds of kinematic waves should be negative on upstream links and positive on downstream links, and (ii) fair merging and First-In-First-Out diverging rules are used to prescribe fluxes from interior states. We prove that, for given initial upstream demands, turning proportions, and downstream supplies, there exists a unique critical demand level satisfying the entropy conditions. It follows that stationary states and kinematic waves on all links exist and are unique, since they are uniquely determined by the critical demand level. For a simple model of urban or freeway intersections with four upstream and four downstream links, we demonstrate that theoretical solutions are consistent with numerical ones from a multi-commodity Cell Transmission Model. In a sense, the proposed theory can be considered as the continuous version of the multi-commodity Cell Transmission Model with fair merging and First-In-First-Out diverging rules. Finally we discuss future research topics along this line.