The classical concept of radiation in gauge theories is analyzed. It is concluded from a discussion of three definitions of electromagnetic radiation, i.e., traditional, Dirac's and Teitelboim's definitions, that only the last of these three represents correctly the structure of electromagnetic self-action. Teitelboim's definition is also satisfactory in the non-Abelian case in which the radiation problem is intertwined with that of confinement. The exact solution of the Yang--Mills equations with current formed by an arbitrarily moving color charge is used as a basis for a description of the non-Abelian classical picture. In the confinement phase, the energy of the gauge field is absorbed by the color charge, whereas the deconfinement phase involves the usual emission of radiation, and the color charge (free or accelerated by non-Yang--Mills forces) produces only colorless converging or diverging waves. Certain other fundamental questions concerning classical self-action in Abelian and non-Abelian gauge theories are also examined.