The successful development of X-ray free-electron lasers (XFELs) as a major scientific instrument is one of the most significant advances in accelerator physics during the last several decades. Central to this development was the accurate prediction of the process in which the initially incoherent radiation from individual electrons evolves to an intense coherent radiation in a long undulator. The calculation involves Klimontovich equation describing the evolution of the electrons’ motion in 6D phase space and 3D Maxwell equation. These coupled equations can be solved in the exponential gain regime by treating the high frequency part of the Klimontovich distribution, which contains electrons’ discreteness and modulations from the FEL interaction, to be small compared to the smooth background. A formal solution can be written down in terms of the Van-Kampen modes, and the resulting equations can then be solved numerically. The excellent agreement of this analysis with the results of elaborate simulation codes gave a solid basis for a practical design of an XFEL.