We present a theory of coupled-bunch longitudinal instabilities that is based on the coupled set of Vlasov equations governing the particle distribution function, and can be applied to arbitrary longitudinal potentials. We find that the coupled-bunch growth rate is given by a dispersion relation that is parametrized by the eigenvalues of the linear (harmonic) coupled-bunch matrix problem. Our theory therefore treats the wakefield-driven source of the instability and the effect of Landau damping together and on equal footing, and also indicates that the stabilizing effects of Landau damping can approximately be compared to the instability growth rates in a harmonic potential that has the same bunch length. We then apply the theory to a weakly nonlinear oscillator and to a quartic potential that is relevant for ultralow emittance storage rings that employ bunch-lengthening systems. In the latter case we find that the theory compares quite favorably with particle tracking simulations for the parameters of the planned upgrade of the Advanced Photon Source.