In this review, basic results of the work on the creation of an RF power supply source for the accelerator with Colliding Linear Electron-Positron Beams (Russian abbreviation: VLEPP) are presented. The work was performed over a period of more than 10 years by teams of the Budker Institute of Nuclear Physics (INP) lab in Novosibirsk and the INP Branch in Protvino, both under the direction of V.E. Balakin. Part of the work was done in collaboration with the Joint Institute for Nuclear Research (JINR) (Dubna). One of the recently manufactured klystron versions was installed and successfully tested at the High Energy Accelerator Research Organization (KEK; Japan).