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

Integration of Superforms and Super-Thom Class