In the upcoming decades, connected vehicles will join regular vehicles to appear on roads, and the characteristics of traffic flow will be changed accordingly. To model the heterogeneous traffic mixing regular and connected vehicles, a generic car-following framework is first proposed in this paper. A linear stability condition is theoretically derived, which indicates that the stability of the heterogeneous traffic is closely related to the penetration rate and the spatial distribution of connected vehicles. The generic car-following framework is applied by taking the Intelligent Driver Model as an example, and it is shown that connected vehicles can obviously enhance the stability of traffic flow and improve traffic efficiency in particular when traffic is in congestion. Moreover, a driver assistance strategy based on distributed feedback control is developed for connected vehicles, and the simulation results show that the proposed driver assistance strategy performs satisfactorily in stabilizing traffic as well as improving traffic efficiency.