The review treats the problem of radiation of relativistic charged particles in matter. The radiation of fast particles in an external field is considered from a unified point of view for an amorphous medium and for a single crystal. The basic attention is paid to the process of radiation in a single crystal where an enhancement of the radiation occurs as compared with an amorphous medium. This effect is shown to be due to the coherent and the interference mechanisms of radiation of relativistic particles in single crystals. First we outline the Born theory (quantum and classical) of coherent radiation of fast particles and show that this theory is valid if the particle propagates through the crystal far from the directions of channeling and if the scattering angle of the particle is small compared with the typical radiation angle of a relativistic particle. Then we show that a violation of these conditions leads to new effects in radiation such as, e.g., the effect of intense radiation of superbarrier and channeled particles, and the effect of suppression of coherent radiation. The comparison of theoretical and experimental results confirms the existence of new effects in the radiation. In conclusion, we review briefly new physical effects that must take place in single crystals at high energies in several other electrodynamical processes.