The original Pizzetti formula describes rotation invariant integration over a sphere as the action of powers of the Laplace operator, showing a close relationship with the theory of spherical harmonics. The goals of this project are to characterize extensions of the Pizzetti formula to the contexts of Stiefel manifolds and Grassmannians and to study the connections with the theory of harmonic polynomials on these homogeneous spaces. By studying rotation invariant integrals over Stiefel manifolds and Grassmannians as the action of rotation invariant differential operators, we aim to provide a powerful tool for efficient computations of complicated integrals appearing in special function, random permutations, and random matrix theory. We plan to uncover the relationship between these integrals and the corresponding theory of harmonic polynomials. These connections will be unraveled by correctly interpreting the Pizzetti integral as a projection operator on a decomposition of the space of polynomials into suitable rotation invariant modules. Moreover, Pizzetti formulae reveal a novel algebraic structure when extended to the supersphere in some exceptional cases for the superdimension, in which there is an exceptional decomposition of super polynomials into harmonic submodules. We will study integration over Stiefel supermanifolds, in the objective of fully understanding these novel structures and their relations with harmonic analysis in superspace.