The Fractional Calculus has been attractive and hot topic among the researchers since 18th century, because of its extensive application in differential and integral equations and other disciplines of mathematics, physics and economics. The motivation of this thesis is to extend the fractional integrals and derivatives, particularly Hadamard fractional integral, and to establish basic properties of the extended fractional integral operators. The application of the extended operators involving the formation of the fractional integral inequalities and solutions of fractional integral equations is focussed in the work. The first chapter includes the introductory background of the fractional calculus. The appropriate literature pertaining to the fractional calculus, involving the theoretical and practical aspects of fractional differential and fractional integral operators has been reviewed. In the second chapter, we have listed symbols, notations and the basic results that are used throughout the dissertation. A number of inequalities involving the Holder’s inequality and AM-GM inequality have been presented. We have defined an extended form of Hadamard fractional integral and have called it Hadamard k-fractional integral. We have also discussed a number of properties of the extended integral operator. In the third chapter, we have established numerous fractional integral inequalities involving the inequalities of Chebyshev functional using the notion of synchronous functions, asynchronous functions, and the like. In the fourth chapter, we have presented some inequalities involving the rearrangement inequalities. On the basis of AM-GM, Holder and the rearrangement inequalities, we have established many fractional integral inequalities related to the extended operator. In the fifth chapter, we have introduced a number of extensions of the fractional integral operators involving the Hadamard type fractional operators. We have discussed the properties of the extended operators involving the semigroup property and commutative law. We have also considered the Mellin transforms and boundedness of some of the extended operators. In the sixth chapter, we have introduced extended fractional derivatives related to the extended fractional integral operators and have discussed their compositions. In the seventh chapter, we have presented some integral equations and have found their solutions using some of the extended fractional integral operators. We have also illustrated the use of some of the extended fractional calculus operators in finding solutions of fractional differential equations.