In this paper a decentralized approach to the platooning problem with nonlinear spacing policies is considered. A predecessor–follower control structure is presented in which a vehicle is responsible for tracking of a desired spacing policy with respect to its predecessor, regardless of the control action of the latter. From the perspective of geometric control theory, we state necessary and sufficient conditions for the existence of decentralized controllers that guarantee tracking and asymptotic stabilization of a general nonlinear spacing policy. Moreover, all nonlinear spacing policies for which there exists a decentralized state feedback controller that achieves asymptotic tracking are characterized. It is shown that string stability is a consequence of the choice of spacing policy and sufficient conditions for a spacing policy to imply string stability are given. As an example, we fully characterize all state feedback controllers that achieve the control goals for a given nonlinear spacing policy, guaranteeing asymptotic tracking for a heterogeneous platoon. The results are illustrated through simulations.