  The non-singlet and singlet evolution kernels of the twist--2 light-ray
operators for unpolarized and polarized deep inelastic scattering are
calculated in $O(\alpha_s)$ for the general case of virtualities $q^2, q'^2
\neq 0$. Special cases as the kernels for the general single-variable evolution
equation and the Altarelli-Parisi and Brodsky-Lepage limits are derived from
these results.
