Braitenberg vehicles have attracted many students to work in robotics because of their apparent simplicity as control mechanisms. However, the lack of a formal theory supporting them entails they are used by the robotic community only as a teaching tool. This paper presents the first joint theoretical and comprehensive analysis of the behaviour of Braitenberg vehicles 2 and 3. The presented mathematical model of the vehicles is a non-linear dynamical system which is analysed for general conditions. This work paves the way to a proper and complete understanding of Braitenberg vehicles through a new theoretical framework. This framework allows the exploration of new applications and shows the need of stimulus analysis to drive its behaviour.