The properties of longitudinal and transverse oscillations in isotropic unmagnetized plasmas of arbitrary composition are investigated on the basis of the Maxwell equations and the relativistic Vlasov equation. We show that in such plasmas (i) superluminal longitudinal and transverse waves undergo no collisionless Landau damping; and (ii) all isotropic particle distributions functions are stable against the excitation of longitudinal and transverse waves. Specializing to the textbook example of longitudinal fluctuations in a pure-electron equilibrium plasma, we derive various analytical expressions for the longitudinal dispersion relation valid for any value of the electron plasma temperature characterized by the dimensionless parameter μ=mec2=kBTe, which allow simpler approximations for non-relativistic (μ[Gt]1) and ultrarelativistic (μ[Lt]1) temperatures. The resulting longitudinal dispersion relation for nonrelativistic temperatures μ[Gt]1 can be expressed in terms of the Fried and Conte plasma dispersion function, but does not agree with the earlier results of Landau, Jackson and others, which were calculated starting from the non-relativistic form of the Vlasov–Maxwell equations. We establish a significant difference in the argument of the plasma dispersion function. This difference only shows up because we have started from a relativistically correct formulation of the kinetic theory, and it profoundly influences the results for the real and imaginary part of the frequency also for non-relativistic plasma temperatures. We show that the non-relativistic limit of the longitudinal dispersion relation reduces to the Landau–Jackson form only if the (unphysical) limit of an infinitely large value of the speed of light c→∞ is taken.