  We address the basic problem of constructing the Thom class for a
supermanifold. Given a cohomological class of a supermanifold and the
restriction of the supermanifold to its bosonic submanifold, the Thom class
gives a prescription to define the integral over the bosonic submanifold in
terms of the integral over the entire supermanifold. In addition, we provide
some new interesting examples of supermanifolds obtained by extending a given
bosonic manifold, we discuss the construction of superforms of special type
(which transform as Berezinian under change of supercoordinates) and we define
the de Rham cohomology. We review the construction of the Thom class in the
conventional geometry and we discuss the extension to the supermanifolds. Then,
we compute explicitly the Thom class for the case of CP^{(1|2)} and, as
expected, the result is singular. We provide a regularization technique to
handle the fermionic Thom class in practical applications. We conclude with
some remarks about Calabi-Yau spaces and their embedding into super-CY.
