  We explicitly compute the first and second cohomology groups of the classical
Lie superalgebras $sl_{m|n}$ and $osp_{2|2n}$ with coefficients in the finite
dimensional irreducible modules and the Kac modules. We also show that the
second cohomology groups of these Lie superalgebras with coefficients in the
respective universal enveloping algebras (under the adjoint action) vanish. The
latter result in particular implies that the universal enveloping algebras
$U(sl_{m|n})$ and $U(osp_{2|2n})$ do not admit any non-trivial formal
deformations of Gerstenhaber type.
