We discuss the progress in the development of extreme light sources that will open new horizons for studying a wide range of fundamental science and astrophysics problems. The regimes of dominant radiation reaction, which completely change the electromagnetic wave–matter interaction, will be revealed, resulting in a new extremely powerful source of ultrashort high-brightness gamma-ray pulses. The possibility of abundant electron–positron pair creation via multi-photon processes and the possibility of reaching the critical field of quantum electrodynamics, which would lead to vacuum polarization and breakdown, are attracting considerable attention.