Fuzzy set theory proposed by Professor L. A. Zadeh is assumed to be generalization of classical or crisp sets. In case of fuzzy sets, it has been accepted that neither the intersection of a fuzzy set with its complement is the null set, nor the union of a fuzzy set with its complement is the universal set. In this regard H. K. Baruah has forwarded an extended definition of fuzzy sets which enables us to define complement of fuzzy set in a way that the axioms of exclusion and contradiction are satisfied by the fuzzy sets also. The sole motivation of this book is to study the notion of fuzzy soft sets in the light of extended definition of fuzzy sets initiated by H. K. Baruah and hence to formulate the fuzziness and softness occurring within the frame work of a number of mathematical avenues. In our work, an attempt has been made to formulate some soft mathematical theories along with a few extensions to some existing frameworks.