Landau damping: half a century with the great discovery

A brief summary is given of the early studies of Landau damping, followed by a discussion of the issues of singularities in the distribution function, reversibility and nonlinear constraints. A difference is emphasized between the evolution of a single-scale localized Langmuir perturbation and a long quasimonochromatic wavetrain. Applicability conditions of the quasilinear approximation are discussed. Examples of the use of the concept of Landau damping in hydrodynamics, astrophysics and other systems are presented.