  We prove the Kobayashi-Hitchin correspondence and the approximate
Kobayashi-Hitchin correspondence for twisted holomorphic vector bundles on
compact K\"ahler manifolds. More precisely, if $X$ is a compact manifold and
$g$ is a Gauduchon metric on $X$, a twisted holomorphic vector bundle on $X$ is
$g-$polystable if and only if it is $g-$Hermite-Einstein, and if $X$ is a
compact K\"ahler manifold and $g$ is a K\"ahler metric on $X$, then a twisted
holomorphic vector bundle on $X$ is $g-$semistable if and only if it is
approximate $g-$Hermite-Einstein.
