Exact solutions of classical gauge theories in even-dimensional ($D=2n$) space–time are discussed. Common and specific properties of these solutions are analyzed for the particular dimensions $D=2$, $D=4$, and $D=6$. The consistent formulation of classical gauge field theories with pointlike charged or colored particles is proposed for $D=6$. The particle Lagrangian must then depend on the acceleration. In $D=2$, radiation is absent and all processes are invertible w.r.t. time. In $D=6$, the expression for the radiation intensity, as well as the equation of motion of a self-interacting particle, is obtained; trembling always leads to radiation. Non-Abelian solutions are absent for any $D\ne4$, and only Coulomb-like solutions, which correspond to the Abelian limit of the $D$-dimensional Yang–Mills–Wong theory, are admitted.