This paper considers the decentralized control of vehicle platoons and gives necessary and sufficient conditions for the existence of such controllers. Specifically, we consider a predecessor–follower control structure 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. By building on geometric control theory, this perspective enables to state necessary and sufficient conditions for the existence of decentralized controllers that guarantee tracking and asymptotic stabilization of a general (linear) spacing policy. For a given spacing policy, these conditions fully characterize the measurements that are required to achieve the desired control objectives. Furthermore, in this geometric framework with the predecessor–follower structure, string stability properties are shown to be the result of the spacing policy rather than the details of control design. As an example, we fully characterize all state and output feedback controllers that achieve the control goals for the constant headway spacing policy. The results are illustrated through simulations.