This dissertation consists of two parts. In the first, we describe the cohomology groups for the subalgebra $\mathfrak{n}^+$ relative to the BBW parabolic subalgebras constructed by D. Grantcharov, N. Grantcharov, Nakano and Wu, essentially with these calculations essentially providing the first steps towards an analogue of Kostant’s theorem for Lie superalgebras. We express these groups in terms of their weight space decompositions relative to the torus, with the weights corresponding to each superalgebra included in the appendix. In the second part, based on joint work with Nakano, we analyze the sheaf cohomology groups $\operatorname{R}^i \ind_B^G L_{\mathfrak{f}}(\lambda)$, where $G$ is a supergroup scheme, $B$ a BBW parabolic subgroup scheme, and $L_{\mathfrak{f}}(\lambda)$ is an irreducible representation for the detecting subalgebra $\mathfrak{f}$. We provide a parametrization of simple $G$-modules and give analogues for the BBW theorem and Kempf’s vanishing theorem for sufficiently large $\lambda$, as well as a criterion for when $\operatorname{soc}_G \operatorname{H}^1(\lambda)$ is simple.