The review is based in a report presented by the author at the RAS Physical Sciences Division's session in honor of Vitaly L Ginzburg's 90th birthday. It examines the current status of theoretical and experimental research on nonlinear phenomena arising when a powerful radio wave propagates in the ionosphere. The focus is on the modification of the ionosphere under the resonance excitation of natural plasma oscillations by radio waves. The upper-hybrid resonance gives rise to strong upper- and lower-hybrid plasma waves; excites strongly elongated ionospheric irregularities, and induces artificial ionospheric radio emission. Nonlinear processes are found to undergo complete transformation near double resonances, when the upper-hybrid frequency is close to a multiple of the electron gyromagnetic frequency. In the neighborhood of the Langmuir resonance, intense plasma waves and ion-sound waves are excited, electrons are effectively accelerated, and an
artificial glow of the ionosphere appears.