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Inequalities Play A Significant Role In All Fields Of Mathematics. Inequalities Are Instru- Mental For The Development Of Many Branches Of Mathematics Such As Functional Analy- Sis, Theory Of Differential And Integral Equations, Interpolation Theory, Harmonic Analysis, Probability Theory, Etc. They Are Also Useful In Mechanics, Physics, And Other Sciences. This Thesis Deals With A Study Of Some Of The Well-Known Inequalities In Classical Math- Ematical Analysis. In This Thesis, Some Important And Useful Inequalities Are Extended To A Wider Class Of Functions. The Work Is Mainly Devoted To Special Kind Of Inequalities, Namely Hardy Type Inequalities For Discrete And Continuous Cases And Opial Type Inequalities For Fractional Order Derivatives. In The Last Three Decades, The Theory Of Hardy Type Inequalities Has Got A Noticeable Development. These Inequalities Played An Important Role For The Development Of Certain Parts Of Mathematical Analysis And Other Branches Of Mathematics As Partial Differential Equations, Function Space Theory, Interpolation Theory, Etc. Hardy-Hilbert’s Inequalities And Reverse Type Hardy-Hilbert Inequalities Are Generalized To L(P) Space. Hardy’s Integral Type Inequalities And Hilbert-Pachpatte Type Integral Inequalities For Fractional Derivatives Are Extended To L(P) Space. The Opial Inequality Involves Integral Of A Function And Its Derivative. Opial’s Work Has Led To Numerous Articles Involving Generalizations, Extensions, And Discretizations. Many Important Applications Of These Inequalities Are In The Theory Of Differential And Difference Equations Such As Establishing Uniqueness Of Solutions Of Boundary Value Problems, And Upper Bounds Of Solutions. Opial Type Inequalities For Fractional Derivatives Are Generalized To L(P) Space. The Results Obtained Are Mainly Theoretical And Can Be Applied For The Characterization Of Other Inequalities.