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The paper comprehensively reviews major methods for analysing local and string stability of car-following (CF) models. Specifically, three types of CF models are considered: basic, time-delayed, and multi-anticipative/cooperative CF models. For each type, notable methods in the literature for analysing its local stability and string stability have been reviewed in detail, including the characteristic equation based method (e.g., root extracting, the root locus method, the Routh–Hurwitz criterion, the Nyquist criterion and the Hopf bifurcation method), Lyapunov criterion, the direct transfer function based method, and the Laplace transform based method. In addition, consistency and applicability of stability criteria obtained using some of these methods are objectively compared with the simulation result from a series of numerical experiments. Finally, issues, challenges, and research needs of CF models’ stability analysis in …