  We study cohomology for classical Lie superalgebras $\mathfrak{g}$ (e.g.
gl(m|n)) over the complex numbers. Using results from invariant theory, we show
that there exist subsuperalgebras which detect the cohomology of
$\mathfrak{g}.$ Furthermore, drawing inspiration from finite groups and
restricted Lie algebras, we introduce the notion of support varieties in this
context. One can prove these varieties have many of the desirable properties of
such a theory. They also appear to provide new interpretations for certain
combinatorics associated with Lie superalgebras.
