The growing social needs in terms of environmental and efficiency issues make the development of automated mobile robots in an off-road context more and more important. Nevertheless, the accurate control of such robots in natural environment requires to take into account several uncertain phenomena, in particular to the varying grip conditions and various delays. If complex models are available to address such problems, their numerous parameters appearing to be variable in an off-road context make them hardly tractable to design an efficient control law. In this paper, an adaptive and predictive control algorithm, based on an extended kinematic model and dedicated to four wheeled steered mobile robots in off-road conditions is proposed for high accurate path tracking applications. The effects of variable low grip conditions are accounted in a kinematic representation thanks to additional variables updated by an observer. This allows the derivation of an adaptive backstepping control approach, able to preserve the accuracy of the path tracking despite of low grip conditions. In addition, a predictive curvature control allows to compensate for the large delays originated from the actuators used on off-road heavy vehicles. The relevance of theoretical developments detailed in this paper are investigated through full scale experiments on both an agricultural tractor (two steering wheels) and a Robucar TT mobile robot (four steering wheels).