  We study the $(k,s)$-positivity for holomorphic vector bundles on compact
complex manifolds. $(0,s)$-positivity is exactly the Demailly $s$-positivity
and a $(k,1)$-positive line bundle is just a $k$-positive line bundle in the
sense of Sommese. In this way we get a unified theory for all kinds of
positivities used for semipositive vector bundles. Several new vanishing
theorems for $(k,s)$-positive vector bundles are proved and the vanishing
theorems for $k$-ample vector bundles on projective algebraic manifolds are
generalized to $k$-positive vector bundles on compact K\"ahler manifolds.
