  We study Chevalley-Eilenberg cohomology of physically relevant Lie
superalgebras related to supersymmetric theories, providing explicit
expressions for their cocycles in terms of their Maurer-Cartan forms. We then
include integral forms in the picture by defining a notion of integral forms
related to a Lie superalgebra. We develop a suitable generalization of
Chevalley-Eilenberg cohomology extended to integral forms and we prove that it
is isomorphic to the ordinary Chevalley-Eilenberg cohomology of the Lie
superalgebra. Next we study equivariant Chevalley-Eilenberg cohomology for
coset superspaces, which plays a crucial role in supergravity and superstring
models. Again, we treat explicitly several examples, providing cocycles'
expressions and revealing a characteristic infinite dimensional cohomology.
