<p>We present a historical review of physical models and computer design methods for high-power klystrons, as well as of original numerical algorithms and computer programs developed for calculations of these devices. The results of the numerical design of a high-power S-band klystron manufactured and tested at the Institute of Nuclear Physics of the Siberian Branch of the Russian Academy of Sciences are described. Klystrons generating a pulse power of tens of megawatts were manufactured in Russia for the first time. They are designed for application as sources of microwave radiation in the fourth-generation synchrotron projects SKIF and the Super c-tau factory. Original algorithms and programs have been developed that allow one to implement the complete klystron design cycle from the cathode to the collector. The comparison of the calculated and experimental data is made. The methods of multidimensional optimization of the main klystron parameters, such as the output radiation power, the gain, and the efficiency, were used successfully.</p>