In modern mobile communication systems, spectral efficient modulation formats have been widely used, which results in the envelope of the radio frequency(RF) signal is variable and having large peak to average power ratio(PAPR). In order to amplify the RF signal without distortion, the linear power amplifier(LPA) is adopted, which is less efficient when powered by constant voltage. Three popular techniques, i.e., Doherty, envelope elimination and restoration(EER), and envelope tracking(ET) techniques, to greatly improve the efficiency of the power amplifiers are analyzed and compared in this paper, and it is shown that the ET technique is the most suitable for future mobile communication systems. In the ET systems, the ET power supply is the key equipment, which dominates the system efficiency. With the development of the mobile communication systems, the bandwidth and the PAPR of the envelope signals are increasing rapidly, which pose severe challenges on the design of the ET power supply. This paper summarizes and sorts the ET power supplies in the literatures, and a detailed comparison is presented to guide the selection of ET power supplies for different applications. Finally, the methods, including soft-switching, slow envelope, and band separation are proposed for further increasing the efficiency of the ET power supply.