  The most general QCD next-to-leading anomalous-dimension matrix of all
possible four-fermion, dimension-six $\Delta F=2$ operators is computed. The
results of this calculation can be used in many phenomenological applications,
among which the most important are those related to theoretical predictions of
$K^0-\bar{K}^0 $ and $B^0-\bar{B}^0$ mixing in several extension of the
Standard Model (Supersymmetry, left-right symmetric, multi-Higgs models, etc.),
to estimates of the $B^0_s$--$\bar B^0_s$ width difference, $\Delta
\Gamma_{B_s}$, and to the calculation of the $O(1/m_b^3)$ corrections for
inclusive b-hadron decay rates.
