The bifurcation analysis of a simple mathematical model describing a road vehicle with a driver is presented. The mechanical model of the car has two degrees of freedom and the related equations of motion contain the nonlinear tyre characteristics. The driver is described by a well-known model proposed in the literature. The road vehicle model has been validated in a case study. Bifurcation analysis is adopted as the proper procedure for analysing both steady-state cornering and straight ahead motion at different speeds. The importance of properly computing steady-state equilibria is highlighted. The effect of a skilled driver is to broaden the basin of attraction of stable equilibria and, in some cases, to stabilise originally unstable behaviours. A subcritical Hopf bifurcation is normally found which limits the forward speed of either understeering or oversteering vehicles. A three-parameter bifurcation analysis is performed to understand the influence on stability of driver gain, of driver prediction time, of vehicle speed. It turns out, as expected from practice, that an oversteering vehicle is more challenging to be controlled than an understeering one. The paper proposes an insight into vehicle-driver interaction. The stabilising or de-stabilising effect of the driver is ultimately explained referring to the existence of a Hopf bifurcation.